Quasi-Linear Parabolic Reaction-Diffusion Systems: A User's Guide
Contents
1. Y E xX gt Eo x 4 W u uo Qu A u u F u u 0 uo Note that u E solves 2 1 on 0 7 with initial value up if and only if u uo 0 0 Consider u as an element of E1 Then Y ux ux 0 0 The assumptions on A and F imply WU C E x X Eo x and DV ux uxu Qu Lxv v 0 v E E From the proof of Theorem 2 1 we know that A u enjoys maximal L regularity The linear operator A ux ux F us is continuous from Xo X1 1 1 pp to Xo i e it is of lower order Thus has maximal L regularity as well see 20 Theorem 6 2 In other words D V uxs ux L E1 Eo x is an isomorphism This gives a neighborhood U of ux in such that ug u uo belongs to C U E 1 where u ug is the solution of 2 1 on 0 7 Moreover for vo X we differentiate U u uo uo 0 in ux to get that Dug tt Ux V9 D1U us us 1 D20 us Ux v0 D1V us ux 0 vo is the unique solution v E1 of uv Lxv 0 on 0 7 with v 0 vo i e Dugul ux e Finally the trace at time 7 is linear and continuous as a map E 4 see 66 Theorem 1 14 5 Applying this to u uo gives the assertion for E Acknowledgments M M and E S thank the CWI for its kind hospitality The authors thank Johannes H wing for his2. 73 G SIMONETT Center manifolds for quasilinear reaction diffusion systems Differential Integral Equations S A K P HE m A 8 1995 pp 753 796 VAN DER STELT A DOELMAN G HEK AND J D M RADEMACHER Rise and fall of periodic patterns for a generalized Klausmeier Gray Scott model J Nonlinear Sci 23 2013 pp 39 95 TRIEBEL Interpolation Theory Function Spaces Differential Operators North Holland Amsterdam 1978 TRIEBEL Theory of Function Spaces II Birkhauser Basel 1983 UECKER Self similar decay of spatially localized perturbations of the Nusselt solution for the inclined film problem Arch Ration Mech Anal 184 2007 pp 401 447 YAGI Abstract Parabolic Evolution Equations and Their Applications Springer Berlin Heidelberg 2010 WU AND X ZHAO The existence and stability of travelling waves with transition layers for some singular cross diffusion systems Phys D 200 2005 pp 325 358 ZUMBRUN Center Stable Manifolds for Quasilinear Parabolic PDE and Conditional Stability of Non classical Viscous Shock Waves preprint Department of Mathematics Indiana University 2008 avail able as arXiv 0811 2788 ZUMBRUN Planar stability criteria for viscous shock waves of systems with real viscosity in Hyperbolic Systems of Balance Laws Lecture Notes in Math 1911 Springer Berlin 2007 pp 229 326 ZUMBRUN AND P HOWARD Pointwise semigroup methods and
3. T u Mul O ul 7 as u gt 0 264 M MEYRIES J D M RADEMACHER AND E SIERO Suppose further that 0 w 0 0mw 0 ker M id Then u 0 is orbitally unstable with respect to E under iterations of T More precisely there is an co gt 0 such that for each gt 0 there are us V with us lt 6 and N EN such that T us E V for n 1 N and dist T us gt 0 Proof We will proceed in four steps Step 1 Let ao 8 gt 0 such that Bsa 0 C V and 4 2 IT uo Muol lt Blluoll lluol lt 5ao There is an approximate eigenvalue re with r gt 1 and 8 R in the spectrum of M Furthermore there are 7 K gt 0 with r n lt r and M lt K r n for all n gt 0 In what follows we choose a 0 o stepwise possibly smaller and smaller depending only on K r n B Y Step 2 Let 6 0 a be given As in the proof of 26 Lemma 5 1 4 we find N N such that a 4 3 lt 6 in N lt 4 3 lt 6 sin NO lt a and u v X with ul 1 and v lt 1 such that 4 4 AZ u iv A u iv lt a n 1 N Here the norm is actually the complexified one i e w1 iw2 w1 w2 for w1 w2 X Define us yu E X such that lus 4 lt 6 Let n 1 N be given Assume inductively that 7 us lt 5ar for k 0 n 1 Then T us is well defined and as in the proof of 26 Theo
4. and the triangular structure of a we conclude that the principle part v gt a wo Urx of F wo is a generator on BUC R RY with domain BUC R R The remaining lower order terms preserve this property The equivalence of the graph norm of F wo and the C norm follows from the 258 M MEYRIES J D M RADEMACHER AND E SIERO boundedness of the coefficients and the open mapping theorem Therefore Theorem 2 6 applies to 2 10 E 2 3 More general problems and other frameworks The above results also hold for smooth dependent nonlinearities provided the principal term a is positive definite uni formly in x Also nonautonomous and nonlocal problems can be treated see 4 5 40 49 Only the mapping properties of the superposition operators and the generator properties of the linearization are relevant Both frameworks cover general quasi linear systems in any di mension if one works with Xo L4 for large q as a base space since then the superposition operators are well defined by Sobolev embeddings Theorem 2 6 also allows us to work in spaces of H lder continuous functions L or subspaces of BUC like Co or C R based on the analytic generator results of 40 and 21 section VI 4 A framework with spatial weights might also be of interest for instance to force some decay of solutions 71 or to treat singular terms 42 Here in particular weights with expo nential growth are straightforward to treat as the generator
5. 160 B SANDSTEDE AND A SCHEEL Relative Morse indices Fredholm indices and group velocities Discrete Contin Dyn Syst Ser A 20 2008 pp 139 158 B SANDSTEDE AND A SCHEEL On the structure of spectra of modulated travelling waves Math Nachr 232 2001 pp 39 93 B SANDSTEDE AND A SCHEEL Defects in oscillatory media Toward a classification SIAM J Appl Dyn Syst 3 2004 pp 1 68 B SANDSTEDE Stability of travelling waves in Handbook of Dynamical Systems II B Fiedler ed Elsevier New York 2002 pp 983 1055 G SCHNEIDER Nonlinear diffusive stability of spatially periodic solutions Abstract theorem and higher space dimensions in Proceedings of the International Conference on Asymptotics in Nonlinear Dif fusive Systems Sendai 1997 Tohoku Mathematical Publishing Sendai Japan 1998 Vol 8 pp 159 167 J SHATAH AND W STRAUSS Spectral condition for instability Contemp Math 255 2000 pp 189 198 E SHEFFER H YIZHAQ M SHACHAK AND E MERON Mechanisms of vegetation ring formation in water limited systems J Theoret Biol 273 2011 pp 138 146 L DE SIMON Un applicazione del la teoria degli integrali singolari al lo studio del le equazioni differenziali lineari astratte del primo ordine Rend Sem Mat Univ Padova 34 1964 pp 205 223 gt gt gt A ADH g ua QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 275 64 65 66 67 68 69 70 71 72
6. 2 1 Well posedness based on maximal L regularity We formulate the results of 33 49 for abstract quasi linear parabolic problems of the form 2 1 ru A uju F u t gt 0 u 0 uo in a Hilbert space setting Let Xo X be Hilbert spaces with X continuously and densely embedded into Xo Roughly speaking Xo is the base space for 2 1 and A u t is an unbounded linear operator on Xo with time independent domain X It turns out that on this abstract level the phase space where the solution semiflow for 2 1 acts is a real interpolation space v Xo X1 1 1 p p DE 1 00 between Xo and X For a definition and the properties of these spaces we refer to the textbooks 11 39 66 At this point we only note that X C C Xo and that is in QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 253 general not a Hilbert space with exceptions for p 2 Fortunately explicit characterizations of X are possible in the concrete settings that we shall use later e g H L H 1 2 9 The real interpolation spaces are the analogues to the fractional power domains in semilinear theory 13 26 These two types of intermediate spaces between Xo and X differ in general again with exceptions for p 2 but are closely related see e g 39 Proposition 4 1 7 Recall from 21 40 that a densely defined operator B on Xo generates a strongly continuous analytic semigroup if and only if A B 2 Xo S uniformly bounded fo
7. 3e 05 2e 05 le 05 0 le 05 2e 05 RA b Figure 3 Spectra of the wavetrains for B C 0 2 D 0 001 A 0 02 before the sideband instability L 5 9 stable near to it L 5 98 and after it L 6 1 unstable a Real part versus linear wavenumber b Imaginary part versus real part In order to link to the formulations for travelling waves let us cast wavetrains as equilibria Wx Vx t W U x ct in the co moving frame x ct with speed c The eigenvalue problem of the linearization of 5 1 in a wavetrain is then given by 5 3 with coefficients of period L 27 k stemming from W The approach via Fourier transform is less useful because the linearization is not di agonal in Fourier space due to the x dependent coefficients As a substitute one uses the Floquet Bloch transform which replaces the eigenvalue problem on R by a family of eigen value problems on the wavelength interval 0 L see section 3 3 Specifically this can be cast as the family of boundary value problems for 0 27 given by 5 3 with 0 replaced by 0 ik L and L periodic boundary conditions With a curve of the spectrum of a wavetrain connected to the origin A 0 due to translation symmetry a change in its curvature is a typical destabilization upon parameter variation This so called sideband instability is illustrated in Figure 3 where we plot spectra of wavetrains in 5 1 passing through a sideban
8. Leuc A respectively It follows from 56 Theorem A 1 that their Fredholm properties are the same as those of Jj 2 A and Tguc A respectively It is further clear that the dimensions of the kernels coincide in both settings Now in 10 Theorem 1 2 it is shown that the Fredholm properties of J 2 A are characterized by exponential dichotomies of the ODE v A A v on both half lines and that in this case the dimension of the kernel of 772 A depends only on the image of the dichotomies This characterization is also true for Tgyc A with the same formula for the dimension of the kernel see 47 Lemma 4 2 and 48 Hence the invertibility and Fredholm properties of J 2 A and Tguc A coincide and if the operators are Fredholm then the dimensions of the kernels coincide This carries over to p2 A and Lguc by the above considerations and shows the assertions a We finally remark that also for the realization of F A on L4 with any 1 lt q lt oo the Fredholm properties are characterized by exponential dichotomies see 10 p 94 Putting this together with the arguments for 56 Theorem A 1 an appropriate generalization of QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 261 Lemma A 3 and interpolation one obtains that the spectrum of is independent of its realization on any of the spaces H 4 and B r where s gt 0 and 1 lt r lt 3 3 Computation of the spectrum The invertibility and Fredholm prop
9. Pattern formation in the one dimensional Gray Scott model Nonlinearity 10 1997 pp 523 563 DOELMAN B SANDSTEDE A SCHEEL AND G SCHNEIDER The dynamics of modulated wave trains Mem Amer Math Soc 199 2009 DORE Mazimal regularity for abstract Cauchy problems Adv Differential Equations 5 2000 pp 293 322 J ENGEL AND R NAGEL One Parameter Semigroups for Linear Evolution Equations Springer Verlag New York 2000 ESCHER J PRUSS AND G SIMONETT A new approach to the regularity of solutions for parabolic equations in Evolution Equations Proceedings in Honor of J A Goldstein s 60th Birthday Marcel Dekker New York 2003 pp 167 190 J W Evans Nerve axon equations IV The stable and the unstable impulse Indiana Univ Math J 24 1974 75 pp 1169 1190 J A GRIEPENTROG AND L RECKE Local existence uniqueness and smooth dependence for nonsmooth T D M 4 zZz g K O quasilinear parabolic problems J Evol Equ 10 2010 pp 341 375 H CKER G SCHNEIDER AND H UECKER Self similar decay to the marginally stable ground state in a model for film flow over inclined wavy bottoms Electron J Differential Equations 2012 2012 61 HENRY Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Math 840 Springer Heidelberg Berlin New York 1981 HOLZER A DOELMAN AND T J KAPER Existence and stability of traveling pulses in a reaction diffusion mechani
10. a steady state of 2 6 and let U CRN be an open neighborhood of the closure of its image For alli j 1 n assume that ci RN is constant that aij U gt RSN and f U RN are C and that aij is positive definite for any U Then for all sufficiently large p 2 00 there is an open neighborhood V of the zero function in Bo a R such that 2 8 is locally well posed in V The solutions belong to H J L N L J H O C J V on time intervals J away from the maximal existence time If U R then one can take V B 2 p Proof The choice Xp L and X H leads to B X Xo X1 1 1 pp for p 1 00 see 66 Remark 2 4 2 4 Let A u v ai u O v and denote by F u the remaining terms on the right hand side of 2 8 The Lipschitz properties of A and F on a neighborhood VY of zero follow from Lemma A 2 For wo V the operator A wo is elliptic the coefficients are bounded and the leading coefficient is uniformly H lder continuous since Bo even embeds into BC for some o gt 0 if n lt 3 and p is large see 66 Theorem 2 8 1 Now the generator property on L follows again from 6 Corollary 9 5 The proof shows that if 2 8 is semilinear then one can apply the results of 13 26 and take a fractional power domain of the Laplacian as a phase space This results in a fractional order Bessel potential space Y H with s sufficiently close to 2 which is still a Hilber
11. comments and are grateful to the anonymous referees for their valuable hints and for pointing out additional references REFERENCES 1 J ALEXANDER R GARDNER AND C K R T JONES A topological invariant arising in the stability analysis of travelling waves J Reine Angew Math 410 1990 pp 167 212 2 H AMANN Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems in Function Spaces Differential Operators and Nonlinear Analysis H J Schmeisser and H Triebel eds Teubner Texte Math 133 Teubner Stuttgart 1993 pp 9 126 3 H AMANN Linear and Quasilinear Parabolic Problems Volume I Abstract Linear Theory Birkhauser Basel 1995 4 H AMANN Nonlocal quasilinear parabolic equations Russian Math Surveys 60 2005 pp 1021 1035 5 H AMANN Quasilinear parabolic problems via maximal regularity Adv Differential Equations 10 2005 pp 1081 1110 6 H Amann M HIEBER AND G SIMONETT Bounded H o calculus for elliptic operators Differential Integral Equations 7 1994 pp 613 653 7 S B ANGENENT Nonlinear analytic semiflows Proc R Soc Edinburgh A 115 1990 pp 91 107 J T BEALE Large time regularity of viscous surface waves Arch Ration Mech Anal 84 1984 pp 307 352 9 M Beck B SANDSTEDE AND K ZUMBRUN Nonlinear stability of time periodic shock waves Arch Ration Mech Anal 196 2010 pp 1011 1076 QUASI LINEAR PARABOLIC R
12. or a front then S is in each setting a family of equilibria of 3 1 Definition 3 1 A pulse or front solution u is called orbitally stable if for gt 0 there is gt 0 such that for ug E X with distx uo S lt 6 the corresponding solution u of 3 1 exists globally in time and satisfies distx u t S lt for allt gt 0 ux is called orbitally stable with asymptotic phase if it is orbitally stable and if for each uo E X sufficiently close to S there is Too such that the corresponding solution of 3 1 converges to U To U as t 4 00 Ux is orbitally unstable if it is not orbitally stable For a wavetrain translates of the profile cannot be realized by localized perturbations Thus only for BUC can orbital stability as above be considered For localized pertur bations i e H or ae stability of a wavetrain is understood with respect to stability of the zero solution of 3 1 3 2 The spectrum of the linearization The linearization of the right hand side of 3 1 in u 0 is 3 2 Ly Afra Bex YY RNxN with smooth coefficients a x G x y x given by a a B d T Tz a T Tr c Oof T Tz y d T Uz Uz a T Tzr Of T Tz Depending on the chosen well posedness framework the operator is considered on Xo H L or BUC with domain H H or BUC where we write Lx for a realization The spectrum of Lx is the set of A C where
13. the corresponding first order operator 0 1 FA BAM ABAL iaa areata which is obtained from rewriting 0 into a first order ODE Hence A z A is a 2N x 2N matrix We write J 2 A and Tguc A for the realization of F A on L R C2 and BUC R C2 respectively with natural domains The following result is rather folklore but does not seem to be explicitly stated in the literature The equality of spectra for realizations on L4 1 lt q lt ow and the space Co of continuous functions vanishing at infinity follows from 52 Corollary 4 6 For the more general theory of dichotomies and spectral mapping results on these spaces we refer to the monograph 12 Proposition 3 2 The following assertions are true where A C e The spectrum the point spectrum and the essential spectrum of Ly Lr2 and Lpuc respectively all coincide e The operator Lr2 X is invertible if and only if Tie is invertible e The operator Lr2 A is Fredholm if and only if Tie is Fredholm In this case the Fredholm indices coincide as well as the dimension of the kernels Proof Lemma A 3 provides an isomorphism T from H to L and from H to H such that Ly T7tLr2T Thus Lyi and Ly2 have for each A C the same invertibility and Fredholm properties It remains to compare 2 and Lguyc A Since a is boundedly invertible these operators have the same invertibility and Fredholm properties as 2 A and
14. ue lao hllz2 lhellz2 C u us llgc llhllz2 IlucllzallAllac lluszllzellhliBc lt C IF C u uz llBc PC u uz llgcllulla2 Pll In the same way we obtain 33 f u ux hella lt C IIF C u Ue lBo IIF C u uz llBcllullz2 llAlla2 Defining F u h Oof u Uz h s f u Uz he we thus have F u Y H H and that ut F u is bounded on bounded subsets of V If h is small then the pointwise identity F u h F u F ujh 1 1 E f f o2f u Tsh ux h Th 033 f U Ug TShz he Thy drds 0 JO and the same types of estimates as above yield F u h F u F u hllan lt CCP h a where C f h is bounded as h 0 These arguments and f 0 0 Ht yield F u H for u V and the differentiability of F in V The Lipschitz property follows from the boundedness of F Iteration for higher derivatives gives F C The arguments apply to u a u on H as well which yields the assertion on A m Note that if f is independent of u then the arguments from the proof above show that f H gt H is smooth Lemma A 2 In the situation of Theorem 2 5 assume in addition that a and f are for some k gt 0 Let A and F be defined by A u u a T u jv F u lai T u O c u u fT u Then for all sufficiently large p gt 2 there is is an open neighborhood V C po of the zero function such that F C V L and A C V Z H L an
15. under abstract conditions a user needs to search for suitable func tion spaces and verify hypotheses that discourage rigor even though some examples provide guidelines However the spectrum of the linearization in a travelling wave can only be meaningfully determined based on a well posedness setting For instance a Turing instability determined via the usual dispersion relation lacks a basis without a consistent phase space Conveniently the pattern forming nature of a Turing instability can be identified ad hoc since the existence of travelling wave patterns is an ODE problem Well posedness is however required to prove that a spectrally unstable solution is indeed unstable under the nonlinear evolution Such a result then justifies the computation of stability boundaries by the spectrum as in 53 65 see also section 5 The purpose of this paper is to present rigorous settings for quasi linear parabolic problems in the travelling wave context as described above We aim for a presentation accessible to practitioners in the spirit of 13 26 59 for semilinear problems To this end we bring together and apply to 1 1 mostly abstract results from the different fields involved in well posedness spectra and stability This puts the naively expected analogy to the semilinear case on firm grounds For quasi linear systems new difficulties mainly arise on a technical level concerning well posedness and nonlinear stability Most importantly a v
16. 0 1991 pp 169 184 NAGEL Towards a matriz theory for unbounded operator matrices Math Z 201 1989 pp 57 68 J PALMER Exponential dichotomies and transversal homoclinic points J Differential Equations 55 1984 pp 225 256 J PALMER Exponential dichotomies and Fredholm operators Proc Amer Math Soc 104 1988 pp 149 156 Pruss Mazimal regularity for evolution equations in Lp spaces Conf Sem Mat Univ Bari 285 2003 pp 1 39 J Pruss G SIMONETT AND R ZACHER On convergence of solutions to equilibria for quasilinear parabolic problems J Differential Equations 246 2009 pp 3902 3931 J Pruss G SIMONETT AND R ZACHER On normal stability for nonlinear parabolic problems Discrete Contin Dynam Systems Supplement 2009 pp 612 621 F RABIGER AND R SCHNAUBELT The spectral mapping theorem for evolution semigroups on spaces of vector valued functions Semigroup Forum 52 1996 pp 225 239 J D M RADEMACHER B SANDSTEDE AND A SCHEEL Computing absolute and essential spectra using continuation Phys D 229 2007 pp 166 183 M R ROUSSEL AND J WANG Transition from self replicating behavior to stationary patterns induced by concentration dependent diffusivities Phys Rev Lett 87 2001 pp 188 302 C J RoussEL AND M R ROUSSEL Reaction diffusion models of development with state dependent chemical diffusion coefficients Progr Biophys Molec Biol 86 2004 pp 113
17. 0 and the smoothness of solutions given by the well posedness it readily follows that 5 1 preserves w gt 0 on the maximal existence interval Assume that u t 2 U x ct is a travelling wave solution of 5 1 with profile u w d BC R R satisfying W gt 6 gt 0 and speed c R Note that this includes homogeneous steady states Denote the co moving frame x ct again by x As for 3 1 the evolution of perturbations u QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 267 of under 5 1 is governed by 5 2 uz a u uz s a uU z s c r uz f U u z Uz Choose V as any open subset of X H X a P with p gt 2 sufficiently large or X BUC such that W w is positive and bounded away from zero for all u w v V This is possible in view of the Sobolev embeddings H C BUC and 2 7 Theorems 2 4 2 5 or 2 7 apply and each yields local well posedness of 5 2 in the corresponding choice of V and in the sense of Theorems 2 1 and 2 7 Solutions are in fact smooth in space and time see Remark 2 3 The eigenvalue problem for the linearization of the right hand side of 5 2 in u 0 is for A E C given by AW Wyr 4Wywy 2Wz w C c w Aw Tw Wu 5 3 2 RES Av Dog coz Bv Tw 20 vv By Proposition 3 2 the spectrum of the linearization is independent of the above functional analytical frameworks A brief account for the computation of t
18. 1 1 p p becomes slightly more complicated to describe It is the N fold product B3 of a Besou space B3 R with s gt 0 and p E 1 00 For s N it follows from 67 Theorem 2 6 1 that an equivalent norm for this space is given by 1 p lulls llull gt J a762 D2u h Dul an jase IMSI where k is the largest integer smaller than s The Besov spaces are closely related to the more common Bessel potential spaces H For any gt 0 we have the dense inclusions H C Bp C H However B H if and only if p 2 and furthermore Bj is a Hilbert space only for p 2 Essential for the applications are the Sobolev embeddings n gt 0 2 7 B3 R CBC R fors gt BS R C LR for s gt 2 wl 3 These are a consequence of B3 C H and the corresponding embeddings for the H spaces For these and many more properties of B spaces we refer to 66 As above we consider a perturbative setting Analogous to 2 4 for perturbations u of a steady state 7 BC R R of 2 6 one is lead to 2 8 Ou O ai a u ju Oi aij u u ju 0 u T fu u 256 M MEYRIES J D M RADEMACHER AND E SIERO Note that the following well posedness result in particular applies to 2 6 when setting 7 0 and assuming f 0 0 Again no symmetry properties of the diffusion coefficients a are required Theorem 2 5 Let n 1 2 3 Let 7 BC R RY be
19. EACTION DIFFUSION SYSTEMS 273 10 11 12 13 14 15 16 Tr 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 A J C J P P BEN ARTZI AND I GOHBERG Dichotomy of systems and invertibility of linear ordinary differential operators Oper Theory Adv Appl 56 1992 pp 90 119 BERGH AND J LOFSTROM Interpolation Spaces An Introduction Springer New York 1976 CHICONE AND Y LATUSHKIN Evolution Semigroups in Dynamical Systems and Differential Equations Math Surveys Monogr 70 AMS Providence RI 1999 CHOLEWA AND T DLOTKO Global Attractors in Abstract Parabolic Problems Cambridge University Press Cambridge UK 2000 CLEMENT AND S Li Abstract parabolic quasilinear equations and applications to a groundwater flow problem Adv Math Sci Appl 3 1994 pp 17 32 CLEMENT AND G SIMONETT Maximal regularity in continuous interpolation spaces and quasilinear parabolic problems J Evol Equ 1 2001 pp 39 67 W CHEN AND M J WARD Oscillatory instabilities and dynamics of multi spike patterns for the one E A A G K J dimensional Gray Scott model European J Appl Math 20 2009 pp 187 214 J DoEDEL AUTO 07P Continuation and Bifurcation Software for Ordinary Differential Equations software 2011 available online at http cmvl cs concordia ca auto DOELMAN T J KAPER AND P ZEGELING
20. HER AND E SIERO 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Y LATUSHKIN J PR SS AND R SCHNAUBELT Center manifolds and dynamics near equilibria of quasi linear parabolic systems with fully nonlinear boundary conditions Discrete Contin Dyn Syst Ser B 9 2008 pp 595 633 J L Lions AND E MAGENES Non Homogeneous Boundary Value Problems and Applications II Springer Berlin 1972 LUNARDI Interpolation Theory Publ Sc Norm Super 9 Springer New York 2009 LUNARDI Analytic Semigroups and Optimal Regularity for Parabolic Problems Progr Nonlinear Dif ferential Equations Appl 16 Birkh user Boston Boston 1995 P K Marini M R MYERscoucGa J D MURRAY AND K H WINTERS Bifurcating spatially heteroge neous solutions in a chemotaxis model for biological pattern formation Bull Math Biol 53 1991 pp 701 719 M MEYRIES Local well posedness and instability of travelling waves in a chemotaxis model Adv Differ ential Equations 16 2011 pp 31 60 MIELKE Instability and stability of rolls in the Swift Hohenberg equation Comm Math Phys 189 1997 pp 829 853 S MORGAN A DOELMAN AND T J KAPER Stationary periodic patterns in the 1D Gray Scott model Methods Appl Anal 7 2000 pp 105 150 NAGA AND T IKEDA Traveling waves in a chemotactic model J Math Biol 3
21. Lx A is not boundedly invertible It is denoted by spec L xo As in the approach surveyed in 28 59 we distinguish between the point spectrum i e E spec Lx such that x A is a Fredholm operator of index zero and its complement within the spectrum called the essential spectrum We will see that point and essential spectrum are independent of the chosen framework and that the familiar spectral theory for ordinary differential operators based on exponential dichotomies as described in 28 59 applies to Usually the set of eigenvalues of Lx is called the point spectrum Note that with the above definition eigenvalues can be contained in the essential spectrum Moreover eigenvalues are not independent of the setting For instance the operator 0 i has a zero eigenvalue 260 M MEYRIES J D M RADEMACHER AND E SIERO with eigenfunction x e on BUC but it is injective on L and Ht Of course this does not contradict Proposition 3 2 on kernel dimensions below since the operator is not Fredholm Since it is assumed that a is positive definite in a neighborhood of the image of the multiplication by a is an isomorphism in each setting Thus the invertibility and Fredholm properties of A are the same as for L A a7 L 2 Ogg 071 80z a Y A which has constant leading order coefficients As before we write Lx A for a realization of L A The key to the spectral properties of L A is
22. SIAM J APPLIED DYNAMICAL SYSTEMS 2014 Society for Industrial and Applied Mathematics Vol 13 No 1 pp 249 275 Quasi Linear Parabolic Reaction Diffusion Systems A User s Guide to Well Posedness Spectra and Stability of Travelling Waves M Meyries J D M Rademacher and E Siero Abstract This paper is concerned with quasi linear parabolic reaction diffusion advection systems on extended domains Frameworks for well posedness in Hilbert spaces and spaces of continuous functions are presented based on known results using maximal regularity It is shown that spectra of travelling waves on the line are meaningfully given by familiar tools for semilinear equations such as dispersion relations and basic connections of spectra to stability and instability are considered In particular a principle of linearized orbital instability for manifolds of equilibria is proven Our goal is to provide easy access for practitioners to these rigorous methods As a guiding example the Gray Scott Klausmeier model for vegetation water interaction is considered in detail Key words quasi linear problem reaction diffusion travelling waves stability orbital instability maximal L regularity Gray Scott Klausmeier model AMS subject classifications Primary 35K57 Secondary 35K59 35C07 35B35 35B36 DOI 10 1137 130925633 1 Introduction In this paper we present rigorous frameworks for well posedness spectra and nonlinear stability of t
23. and fronts Recall the precise notion of orbital stability from Definition 3 1 An application of 50 51 gives the following conditional result For more infor mation on semisimple eigenvalues in Banach spaces we refer the reader to 40 Appendix A 2 Proposition 4 1 Let U have constant asymptotic states Assume that A 0 is a semisimple eigenvalue of L with eigenfunction UW i e ker L span u and Xo ker L imL As sume further that the remaining part of spec L is strictly contained in ReX lt 0 Then the travelling wave ux is orbitally stable with asymptotic phase and limit translates u Too are approached exponentially Proof By translation invariance it suffices to consider S in a neighborhood of r 0 The framework of Theorem 2 1 is that of 50 Theorem 2 1 provided that in addition A and F belong to Ct which is guaranteed by the assumption on a and f The setting of Theorem 2 6 is that of 51 Example 3 To apply 50 Theorem 2 1 and 51 Theorem 3 1 it remains to verify that zero is normally stable in the sense of 50 51 We have that S is a one dimensional C manifold with tangent space at r 0 spanned by U By assumption the tangent space QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 263 coincides with the kernel of and zero is a semisimple eigenvalue Hence normal stability follows Oo For a quasi linear variant of the Huxley equation the above conditions have been verified in 50 sect
24. ariation of constants formula is not available Further when dealing with quasi linear problems one has to take into account all available regularity as prescribed by sharp trace results such that in general one cannot take fractional power domains as a phase space for the solution semiflow Instead one has to work with real interpolation spaces see section 2 1 or the domain of the linearized operator itself However in the end it turns out that the familiar spaces H and BUC are possible phase spaces and that the spectral theory and the sufficient conditions for nonlinear stability are analogous to the semilinear case at least in noncritical cases There are several abstract settings for well posedness of general quasi linear parabolic problems available in the literature see 2 5 15 24 29 33 40 49 69 and 4 as well as section 2 3 for a selective overview These have advantages and disadvantages depending on the present context and the geometric qualitative theory is more or less developed in each case On the other hand solutions may be constructed by fixed point arguments tailor made for the issues under investigation e g 71 The real viscous conservation laws are an important and well studied class of quasi linear problems where well posedness results exploit the additional QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 251 structure 31 We refer to the survey 72 and the references therein Our focus lies on the app
25. cations In a smooth setting the equation uz a u Ucr f u uz can be cast into divergence form by a suitable redefinition of a and f We believe that also the more general results in 57 on spectra of modulated travelling waves carry over to the quasi linear case but we do not enter into details here Also the nonlinear stability of wavetrains is not considered This is a delicate issue since zero always lies in the essential spectrum Hence the best one can hope for is heat equation like decay 252 M MEYRIES J D M RADEMACHER AND E SIERO Under certain assumptions this has been established for the semilinear reaction diffusion case in 19 60 A special quasi linear case more precisely the quasi linear integral boundary layer model is considered in 25 Also for viscous shocks the spectrum touches the origin and stability in weighted spaces can be established We refer to 73 the survey 72 and the references therein as well as to 9 for more recent results In section 5 we illustrate our general considerations by means of the Gray Scott Klausmeier vegetation water interaction model 32 for x R given by wi w ze Cwr A 1 w wv 1 2 a vi Doge Bu wv with constants A B gt 0 C R and D gt 0 This system is the original motivation for the present study It is quasi linear due to the porous medium term w ss 2 wwWee wz and is therefore parabolic only in the regime
26. cs system J Nonlinear Sci 23 2013 pp 129 177 KAPITULA AND K PROMISLOW Spectral and Dynamical Stability of Nonlinear Waves Appl Math Sci 185 Springer New York 2013 KATO Quasi linear equations of evolution with applications to partial differential equations in Spectral Theory and Differential Equations Lecture Notes in Math 448 Springer Verlag Berlin 1975 pp 25 70 KATO Perturbation Theory for Linear Operators 2nd ed Springer Berlin Heidelberg New York 1980 KAWASHIMA Large time behavior of solutions for hyperbolic parabolic systems of conservation laws Proc Japan Acad Ser A 62 1986 pp 285 287 A KLAUSMEIER Regular and irregular patterns in semi arid vegetation Science 284 1999 pp 1826 1828 KOHNE J PRUSS AND M WILKE On quasilinear parabolic evolution equations in weighted Lp spaces J Evol Equ 10 2010 pp 443 463 KUMAR AND W HORSTHEMKE Turing bifurcation in a reaction diffusion system with density dependent dispersal Phys A 389 2010 pp 1812 1818 LADYZHENSKAYA V SOLONNIKOV AND N URAL TSEVA Linear and Quasi linear Equations of Parabolic Type Transl Math Monogr Amer Math Soc Providence RI 1968 LATUSHKIN J PRUSS AND R SCHNAUBELT Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions J Evol Equ 6 2006 pp 537 576 274 M MEYRIES J D M RADEMAC
27. d both maps are Lipschitz on bounded sets One can take V Bo if U RY Proof Since n lt 3 from Sobolev s embedding 2 7 we find p gt 2 such that B C H4 ABC Then V can be chosen such that the image of 7 u is strictly contained in U uniformly in u V The regularity of A and F can be derived as in Lemma A 1 using F 0 0 The need for po c H and thus also H C H comes from the nonlinear 2 2 p 2 p Crt gradient terms Indeed assume for simplicity that w 0 Then for u1 u2 E B and v H we can estimate lat u1 i u1 jv al u2 iuz jvllz2 lt laly ur Oierr a4 2 Oye 419 04 lt lla ua feller uzlina luall ella u1 aiz u2 llBc lvl za employing H lder s inequality L4 L4 C L in the first line E QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 271 A 2 A commuting isomorphism for elliptic operators The following auxiliary result for second order differential operators allows us to transfer spectral properties from L to H by conjugation Lemma A 3 Let a 8 y BC R RN and assume that a x is positive definite uni formly in x Then there is a continuous isomorphism T H L which also maps T H gt H isomorphically that commutes on H with the operator p gt LY apra Boy YY Proof The isomorphism T will be the square root of a shift of The main point is to show that its domain for the realization on H is H Denot
28. d instability as the wavelength L changes For these computations we implemented the first order formulation of the dispersion relation numerically in AUTO based on the algorithm from 53 As for the homogeneous steady state the wavetrains with unstable spectrum e g L 6 1 in Figure 3 are expected to be orbitally unstable under the nonlinear evolution of 5 1 see Proposition 4 2 and Theorem 4 3 Appendix A Auxiliary results A 1 Superposition operators We give some details for the properties of the nonlinear maps employed in the well posedness results Lemma A 1 Let U1 U2 C RY be open neighborhoods of zero let a R x U gt RN be C 8 and let f Rx U x Uz gt RY be C with f 0 0 H Define the superposition operators A u u a U Ux ars F u f u ux Then there is an open subset V of H such that A C V 2 H H and F C V H and both maps are Lipschitz on bounded subsets of V One can take V H if Uy Uz R AtTE V the derivatives for u H and v H are given by A T u v O2a T fu ve F uov Oof U Tx V O3f T Ue ve 270 M MEYRIES J D M RADEMACHER AND E SIERO Proof Choose V C H such that for u V the closure of the images of u uy H c BC are uniformly contained in U and Us respectively Let u V For h H we use juhl z2 lt llullec Al z2 and lullac lt C ul g to estimate l2 f u Ue Allzn lt f C u
29. dent of B 0 Since 0 has full rank m we have Cy minjg 0 gt 0 and we can choose 79 such that CpTo lt Cy 2 Hence with V 20 Cy and small a we obtain MOI WOXI Clc gt 10a for 7 gt gt Va Then with these choices dist T us inf T us YO I lt va Step 4 Now let lt Ya Then 4 5 4 8 and the estimates 4 6 and sin NV lt a yield IT us HOI AX us WOLI El lEn lle 4 9 gt jaeN u 0 8a Cura 0 Cyna The vectors u and w 0 are linearly independent if is sufficiently small In fact otherwise our assumption w 0 ker M id would imply that Mu u But as in 4 7 the estimate 4 4 then yields A 1 Au Mull lt a 2a which is impossible for small a We conclude that u 4 0 is bounded away from zero uniformly for lt Ya Hence decreasing a once more if necessary we obtain from 4 9 and o gt 1 that dist T us gt o where o gt 0 is a multiple of a independent of a Let us now apply the lemma to abstract quasi linear problems 4 10 ru A uju F u t gt 0 u 0 uo We denote by L ux A us A us ux F us the linearization of the right hand side at Ux Theorem 4 5 Assume the setting of either Theorem 2 1 or 2 6 and in addition that A and F are C Let E C VN X be an m dimensional C manifold of
30. e by L 2 the realization of on L with domain H The properties of a together with 6 Theorem 9 6 imply that there is some w gt 0 such that B w 72 is a negative sectorial operator and has a bounded holomorphic functional calculus of angle strictly smaller than gt In particular T B is a well defined continuous isomorphism D B gt L see 66 Theorem 1 15 2 The boundedness of the holomorphic calculus of B implies that it has the property of bounded imaginary powers Therefore combining 39 Lemma 4 1 11 with 66 Theorem 1 15 3 or 39 Theorem 4 2 6 we have D B L H 1 2 where Jag denotes complex interpolation see 11 39 66 Since L H Ht by 66 Remark 2 4 2 2 it follows that T H L is an isomorphism Next we show that T H gt H is an isomorphism Again by 66 Theorem 1 15 2 T also maps isomorphically D B gt D B H We show that D B H as Banach spaces By 39 Lemma 4 1 16 and Theorem 4 1 11 and the previous considerations we have D B3 u D B Bu D BY u H Lu Ht For u H we clearly have Lu H and hence H C D B 2 Conversely let u H such that Lu Ht Then auser Y Buz yu Lu H By assumption the coefficient a is pointwise invertible with a BC Therefore uz a H and so u H We conclude that D B3 2 H as sets Arguing as before we get lull ogs llu
31. e problem formulation is equivalent to L Oz V AV V 0 e V L By Floquet theory this precisely means that the period map II A of the evolution operator for the ODE L 0 U AU possesses an eigenvalue a Floquet multiplier e Hence also here a linear dispersion relation can be defined by d A det TI A e 0 which precisely characterizes the spectrum An important difference from the case of ho mogeneous steady states is that A 0 always lies in the essential spectrum x independent coefficients of 3 1 yield a trivial zero Floquet exponent which implies that d 0 0 0 Indeed B 0 0 in this translation symmetric case Finally in case of a generalized wave train the boundary of the essential spectrum of is as above obtained by replacing the coefficients of with its periodic limits at oo and considering the dispersion relation The point spectrum is also given by an Evans function see 58 section 4 there also the more general case of time periodic solutions so called defects is treated 4 Nonlinear stability and instability For the nonlinearities a f and a travelling wave solution u t U a ct of 1 1 we make the same assumptions as in the previous section We consider 3 1 Ut a Vise Neg a Ub ee Ves c t Ur fu H U Uy 4 ugr in any of the well posedness settings in a neighborhood of S u 7 U 7 R 4 1 Stability of pulses
32. equilibria of 4 10 parametrized by w U C R gt E and let u E E satisfy e spec L u N Re gt 0 9 OW Ca OmY Ca ker L usx for ux U G Then u is orbitally unstable in V C X with respect to E Proof Shrink V around u if necessary such that t ug gt 1 for each ug V Let V be the time one solution map for 4 10 Define T uo 1 us uo ux uo 266 M MEYRIES J D M RADEMACHER AND E SIERO for uo close to u Then T is continuous T u 0 for u ENY and T satisfies 4 1 with M E0 E Z X as a consequence of Proposition A 4 for the setting of Theorem 2 1 and of 42 Proposition 6 2 for the setting of Theorem 2 6 Moreover M has spectral radius larger than one by 40 Corollary 2 3 7 and interpolation and 0 7 ker M id follows from the assumption Thus Lemma 4 4 applies Oo Of course Lemma 4 4 applies in any well posedness setting for nonlinear parabolic prob lems 5 A generalized Gray Scott Klausmeier model For illustration of the previous results let us consider the model 1 2 for water vegetation interaction in semiarid landscapes wi w xe Cwr A 1 w wr vi Dovey Bu wr 5 1 Here A is roughly a measure of the rainfall On the one hand 5 1 is a rescaling of the Klausmeier model for banded vegetation patterns on a sloped terrain from 32 when removing the porous medium term w On the ot
33. er trick see 49 Theorem 5 1 and 22 one can show that for smooth nonlinearities the solutions of 2 2 are smooth in space and time When investigating the stability of a nonlocalized travelling wave with respect to localized perturbations one is led to a variant of 2 2 with x dependent nonlinearities Furthermore in many situations the nonlinearities are not everywhere defined on RY or the leading coefficient a is positive definite only in a subset of R For instance this is the case for the Gray Scott Klausmeier model 1 2 where the focus lies on perturbations of travelling wave solutions in the parabolic regime w gt 0 For a general formulation let 7 BC R R be a steady state of 2 2 i e 2 3 a U te e f U Ur 0 Then u solves 2 2 for a perturbation u if and only if u solves 2 4 Ut a T U Uz e A U Utz ae f T U Ue Uz For this perturbative setting we have the following variant of Theorem 2 2 Here and in the following the image of is meant to be the set u x x R Theorem 2 4 Lett BC R RY satisfy 2 3 and let U1 U2 C RY be open neighborhoods of the closure of the images of resp Tz Assume that a U RN is C such that a C is positive definite for any C U1 and f Ui x Uz gt RN is O QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 255 Then there is an open neighborhood V of the zero function in H such that 2 4 is locally we
34. erties of F A and thus the characterization of the point and essential spectrum of are described in terms of exponential dichotomies in 59 section 3 4 This is independent of the variable leading order coefficients of due to its quasi linear origin and thus the same as for semilinear reaction diffusion systems We briefly describe the main points for each type of wave A detailed discussion can also be found in 28 Chapter 3 For a homogeneous steady state the point spectrum of the constant coefficient operator is empty Since the Fourier transform is an isomorphism on L the essential spectrum can be determined by transforming to Llr ar ibr 7 E CNN KER Now we have spec if and only if d K det L K A det A A ix 0 for some which is called the dispersion relation for The latter also means that A A is a nonhyperbolic matrix Thus here it is straightforward to determine the spectrum at least for N not too large For pulses and fronts replacing the variable coefficients of by their values at 00 leads to constant coefficient operators L whose spectrum is determined as just described For pulses the essential spectrum of already coincides with spec For fronts spec equals the boundary of the essential spectrum of which usually already determines stability issues This is related to the fact that replacement by the values at infinity is a relatively compact pertu
35. he spectrum is given in section 3 3 and we refer to 59 for a survey Nonlinear stability or instability of u can be deduced from the results in section 4 in some situations as pointed out below 5 2 Homogeneous steady states These are solutions w t x wx v t z v E R to 5 1 that are time and space independent and thus solve the algebraic equations arising from vanishing space and time derivatives We readily compute that the possibilities are wo vo 1 0 and in case A gt 4B i 4 4 4f A2 4AB E T 4 4f A2 4AB l 2B The state wo vo with zero vegetation represents the desert even though there is nonzero water while the equilibria w v and w_ v_ represent co existing homogeneously vegetated states At A Asn 4B the latter two collapse in a saddle node bifurcation The spectrum of the linearization in w v can be computed from the usual dispersion relation d A x 0 where B 2w K ik C c A v 22W Ux ahde v2 Dx ike B 2u 0 is obtained from a Fourier transform see section 3 3 An origin of patterns is a supercritical Turing Hopf bifurcation of the steady state w4 v4 that occurs as A decreases from larger values as shown in 65 It is in fact straight forward to study bifurcations of spatially periodic travelling waves as this involves only ODE analysis As a side note on Turing Hopf bifurcations we mention that the dynamics
36. he upper and lower w 1 branches b Magnification of the bifurcation diagram with bullet marking the location of the sideband instability at L 5 98 Solutions on the branch for increasing period are spectrally stable numerically using AUTO 17 from the dispersion relation as the parameter A passes through the aforementioned Turing Hopf bifurcation Since the spectrum is unstable after passing the Turing Hopf instability e g A 0 43 in Figure 1 the steady state is expected to be unstable under the nonlinear evolution Indeed this is the case thanks to Theorem 4 3 5 3 Wavetrains The patterns emerging at the Turing Hopf bifurcation are periodic wavetrains which are solutions to 5 1 of the form ws V t 0 kx wt with a 27 periodic profile w 0 Here w is called the frequency and k the wavenumber As noted in 65 the existence region of wavetrains to 5 1 in parameter space extends far from the Turing Hopf bifurcation and even beyond the saddle node bifurcation A Asn of homogeneous equilibria with vegetation In Figure 2 we plot a branch of wavetrain solutions for A lt Asn that appears to terminate in another type of travelling wave pulses which are spatially homoclinic orbits QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 269 2e 05 1e 05 4 ot 1e 05 R A 2e 05 H 3e 05 4e 05 5e 05 F 6e 05 aoi pee o a eg 3 6e 05 5e 05 4e 05
37. her hand upon replacing w 2 by wrx and setting C 0 5 1 is precisely the semilinear Gray Scott model which has been extensively studied in the past decades see e g 16 18 44 and the references therein The relations between these different models in terms of periodic patterns have been studied in 65 From an application point of view it is important to know in which patterned state these model systems may reside and thus to establish well posedness as well as existence stability and instability of patterns In order to illustrate the straightforward applicability of the frameworks of the previous sections we show well posedness around travelling waves with first component bounded away from zero We then consider homogeneous steady states and wavetrains and derive the dis persion relations These are illustrated by numerical computations of spectra when passing a Turing Hopf bifurcation and a sideband instability 5 1 Well posedness for perturbations of travelling waves To cast 5 1 into the form 1 1 we set u w v and define the smooth nonlinearities a R R and f R R by _ 2w 0 _ Cuz Al w wv a u 0 D f u Ur Bu wv2 Then 5 1 is equivalent to uz a u u z f u uz We see that a u is positive definite only for w gt 0 and thus 5 1 fails to be parabolic for w lt 0 We therefore restrict our analysis to w gt 0 From the quasi positive structure of f for A gt
38. ik Postfach 33 04 40 28359 Bremen Germany rademach math uni bremen de This author s work was partially supported by the NWO cluster NDNS and by his previous employer Centrum Wiskunde amp Informatica CWI Amsterdam Mathematisch Instituut Universiteit Leiden P O Box 9512 2300 RA Leiden The Netherlands esiero math leidenuniv nl 249 250 M MEYRIES J D M RADEMACHER AND E SIERO the simplest interesting reaction diffusion patterns and are observed for different types of quasi linear systems see e g 27 34 41 42 45 62 70 For semilinear parabolic problems on the line it is well known that e g H or BUC are suitable phase spaces for well posedness in a perturbative setting 13 26 The corresponding spectrum of the linearization is characterized in terms of the dispersion relation and the Evans function 23 59 In some situations in particular when the essential spectrum does not touch the imaginary axis nonlinear orbital stability of a wave can directly be deduced by a principle of linearized stability 26 60 An excellent reference for the spectrum and stability of nonlinear waves in the semilinear context is 28 For quasi linear models an analogous unified framework for well posedness spectra and stability of waves seems less known It seems that the majority of concrete well posedness results in the literature concern bounded domains Moreover when the general results are formulated abstractly or
39. ing wave and other pattern type solutions is the Sobolev space H Theorem 2 4 For nonlocalized perturbations BUC C functions bounded and uniformly continuous with all derivatives can be chosen as a phase space Theorem 2 7 e For arbitrary space dimensions x R the higher order Sobolev spaces H with k gt 1 are still possible phase spaces see section 2 1 2 For n lt 3 one can take certain Besov spaces real interpolating between L and H Theorem 2 5 In the latter case the advantage is that the linearization can be directly considered on L e The spatial dynamics spectral theory developed for semilinear parabolic systems on the line applies also in the quasi linear case which allows us to compute the spectrum of travelling waves in a familiar way see section 3 3 In particular the spectrum is independent of the chosen setting Proposition 3 2 e The well known nonlinear stability result with asymptotic phase for travelling waves with simple zero eigenvalue applies in these settings Proposition 4 1 as a direct consequence of 50 51 e Without assuming a spectral gap or an unstable eigenvalue it is shown that an unstable spectrum implies orbital instability of pulses and fronts Theorem 4 3 and instability of wavetrains Proposition 4 2 Here we rely on a general result on orbital instability of manifolds of equilibria Lemma 4 4 We emphasize that the divergence form 1 1 is assumed only in view of appli
40. ion 5 by elementary arguments An abstract and more general variant of Proposition 4 1 and applications to semilinear problems can be found in 28 Chapter 4 4 2 Instability of generalized pulses ene am under localized perturbations For lo calized perturbations i e for H or Bo 2p P a generalized pulse or front u is nonlinearly stable or unstable if the zero solution of 3 1 is stable or unstable as a single equilibrium in the sense of Lyapunov Nonlinear stability is a delicate issue see the discussion in the introduction In case of an unstable spectral value we have the following Proposition 4 2 If has a periodic asymptotic state and spec N ReA gt 0 0 then the generalized front or pulse u is nonlinearly unstable with respect to localized perturbations from X H or X B 2 P Proof The Lemmas A 1 and A 2 together with Proposition A 4 imply that the time one solution map for 3 1 obtained in Theorems 2 4 and 2 5 from Theorem 2 1 is C around zero with 0 ef Y X Considered on 2 Xo this operator has spectral radius larger than one by 40 Corollary 2 3 7 Using w with sufficiently large w gt 0 as a conjugate this property carries over to ef considered on Y X Now it follows from interpolation that the realization of ef on Y X has spectral radius greater than one Thus the zero solution of 3 1 is unstable by 26 Theorem 5 1 5 a 4 3 Orbital instability Without assuming a s
41. l C norm It is shown in 40 that a scalar second order elliptic operator on BUC BUC behaves well and generates an analytic semigroup This is the main ingredient for applying Theorem 2 6 as follows The triangular structure of a is assumed for simplicity Theorem 2 7 Let BUC R R be a steady state of 2 10 and let U1 Uz C RY be open neighborhoods of the closure of image of resp Uz Assume that a Uy RYN and f U x Uz gt RY are C such that e for each U the matrix a is triangular and the diagonal entries of a are positive and bounded away from zero uniformly Then there is an open neighborhood V of in BUC such that 2 10 is locally well posed in V One can take V BUC if U Uz RN Proof Choose an open set V C RY that contains the image of and satisfies V C U Define V as the set of all w BUC with image contained in V Then F u a u uz z f u uz defines a superposition operator F V BUC It is straightforward to check that F C V BUC At wo V we have F wo v a wo vx e a wo wo e Vl e cvr f wo r v BUC and F V gt Z BUC BUC is locally Lipschitz For the generator property let wo V be given By 40 Corollary 3 1 9 each of the scalar valued operators v gt ai wo Wre with domain BUC generates an analytic Co semigroup on BUC where aj are for i 1 N the diagonal entries of a Using the matrix generator result 46 Corollary 3 3
42. ll posed in V If U1 Uz RN then one can take V H Proof Let again Xo H X H and p 2 such that H Define 2 5 A u v aU u vz e F u a 4 ute f T u Ge Uz Using F 0 0 Lemma A 1 yields V C H such that F V gt H and A V gt H H are Lipschitz on bounded sets If V is sufficiently small then for each w V the leading coefficient a wo of A wo is positive definite uniformly in x R Thus as in the proof of Theorem 2 2 it follows from 6 Corollary 9 5 and an interpolation argument that A wo with domain H has the required generator property on Ht to apply Theorem 2 1 Oo 2 1 2 Well posedness in space dimensions n lt 3 For simplicity on R we consider quasi linear reaction diffusion advection problems using the sum convention 2 6 ut O aij u Oj u GOju f u xz e R Here essentially aj RY Rs ee RYN fori j 1 n and f RY gt R The approach of the previous subsection works in any dimension if one takes Xo HE R RY with k gt 5 as a base space since then H k is an algebra and the superposi tion operators are Lipschitz as before This leads to the Hilbert space H as a phase space We present another functional analytic setting with X L as a base space for which Theorem 2 1 applies to 2 6 in space dimensions n lt 3 The price one has to pay in the maximal L regularity approach is that the phase space L H
43. lla ull lt Cull for a constant C that is independent of u Since we already know that H is complete with respect to 5 3 2 and z3 the converse estimate follows from the open mapping theorem Finally it follows from 39 Theorem 4 1 6 that w 2 and its square root T commute on H This implies that also 2 commutes with T Oo The assertion of the above lemma remains valid with literally the same proof if one replaces the L setting by an L4 setting where q 1 00 A 3 The time one solution map We use the implicit function theorem to prove that in the neighborhood of an equilibrium the solution semiflow obtained from Theorem 2 1 for 2 1 is as smooth as the right hand side See 26 Theorem 3 4 4 for the semilinear case as well as 40 Theorem 8 3 4 and 5 Theorem 4 1 for quasi linear frameworks Proposition A 4 In the situation of Theorem 2 1 assume additionally that ASOW 2G FEC V Xo for some k EN Let uz E VAX be an equilibrium of 2 1 i e A us us F usx 0 Then for any T gt 0 there is a neighborhood U C V of ux such that the time r map uo gt ug 272 M MEYRIES J D M RADEMACHER AND E SIERO u T uo for the solution semiflow for 2 1 is well defined and belongs to C U X Moreover let Ly A us A us Ux F us Then us e Proof We assume V Set Ey H 0 7 Xo N L 0 7 X1 and Eo L 0 7 Xo and consider
44. of 5 1 near onset is formally approximated by a complex Ginzburg Landau equation see 65 but a rigorous justification has not been established for quasi linear problems to our knowledge In order to locate the Turing Hopf bifurcation we need to study the spectrum of the lin earization in w v For illustration in Figure 1 we plot the spectrum obtained 268 M MEYRIES J D M RADEMACHER AND E SIERO 0 10 0 030 0 020 0 010 0 000 0 010 0 020 0 030 0 30 0 25 0 20 0 15 0 10 0 05 0 00 0 05 0 10 RA b Figure 1 Spectra of the homogeneous steady state wt v of 5 1 for B C 0 2 D 0 001 before the Turing Hopf instability A 0 63 stable near to it A 0 53 and after it A 0 43 unstable a Real part of the spectrum versus linear wavenumber b Imaginary part of the spectrum versus real part 0 017 0 00070 0 015 0 00060 0 012 il 0 00050 0 010 0 00040 gt 0 008 w 0 5 0 00030 0 005 0 00020 0 003 0 0 00010 0 000 0 L 0 00000 0 003 i 0 00010 r e i 0 25 50 15 100 3 0 3 5 4 0 4 5 50 55 6 0 6 5 7 0 a b Figure 2 a Sample bifurcation diagram of wavetrains for A 0 02 B C 0 2 D 0 001 At L 3 45 a fold occurs and both branches appear to terminate in a homoclinic bifurcation as L co The inset shows profiles of solutions at the fold w 0 5 and near L 80 on t
45. oned the phase space is now a subset of X and not of an intermediate space between Xo and X1 Well posedness is similar to that of Theorem 2 1 The maximal existence time is lower semicontinuous and the solution semiflow is continuous with values in V For each a 0 1 and an initial value uo V one obtains a unique maximal solution u of 2 9 such that u BUC1 0 T Xo NBUC2 0 T X1 for T lt t ug Here BUC is a weighted H lder space see 3 Chapter II 2 and 51 Example 3 It is slightly confusing that these spaces differ from the ones in 40 denoted by C but BUC is indeed the regularity obtained in 40 Theorem 8 1 1 Theorem 2 6 applies to 2 2 2 4 and 2 8 under assumptions similar to those of Theo rems 2 2 2 4 and 2 5 with different phase spaces In particular instead of a Besov space one obtains H as a phase space in the setting of Theorem 2 5 We do not formulate the precise results but rather consider a setting for reaction diffusion systems which is not covered by the approach of Theorem 2 1 2 2 1 One space dimension Well posedness in BUC We reconsider the case of one space dimension i e for u t RY the problem 2 10 ut a u uz e f u us t gt 0 TER We present a setting in which nonlocalized perturbations of steady states can be treated For k No denote by BUC BUC R R the Banach space of bounded uniformly continuous functions endowed with the usua
46. pectral gap or the existence of an unstable eigenvalue we show that an unstable spectrum implies orbital instability Theorem 4 3 The following assertions are true e Let have constant asymptotic states Assume spec L Rev gt 0 0 Then ux is orbitally unstable with respect to localized and nonlocalized perturbations from X H By or X BUC Let U have a periodic asymptotic state Assume spec L N Re gt 0 0 Then ux is orbitally unstable with respect to nonlocalized perturbations from X BUC This result is a direct consequence of the general orbital instability result Theorem 4 5 below for manifolds of equilibria W X4 in the settings under consideration and Lu 0 by the exponential convergence of at infinity and translation invariance of the equation The following lemma and its proof are generalizations of 26 Theorem 5 1 5 and 61 Similar to those results the proof establishes that perturbations of suitable approximate unstable eigenfunctions deviate from the manifold of equilibria Lemma 4 4 Let X be a real Banach space let V C X be an open neighborhood of zero and let E C V be an m dimensional C manifold containing zero Let E be parametrized by an injective map Y U CR E with w 0 0 where y 0 has full rank m Assume that T YV gt X is continuous that T u 0 for u E and that there is an M LY X with spectral radius greater than one such that for some c gt 1 4 1
47. r A from a left open sector in C As a consequence of the results in 33 49 we have the following Theorem 2 1 Let p 1 00 and X C X C Xo be as above Assume that there is an open set VC X such that e F V gt Xo and A V gt L X4 Xo are Lipschitz on bounded sets e for each wo E V the operator A wo with domain X generates a strongly continuous analytic semigroup on Xo Then 2 1 is locally well posed in V with solutions in a strong LP sense More precisely the theorem yields solvability of 2 1 as follows For each initial value ug V there is a maximal existence time t uo gt 0 and a unique solution u u u0 C 0 t uo V of 2 1 such that u H J Xo N LP J X1 for time intervals J 0 T with T lt t uo Here H J Xo denotes a vector valued Sobolev space which is defined as in the scalar valued case An equivalent norm on H J Xo N L J X1 is given by Ce f eOe lutt dt Furthermore t uo is finite only if either dist u t uo V 0 or u t uo l x gt oo as t t uo The map tt V 0 00 is lower semicontinuous and the local solution semiflow t uo gt u t uo is continuous with values in V C If F and A are smooth then the semiflow enjoys smoothness properties as well We demonstrate this in Proposition A 4 in the appendix for a neighborhood of a steady state Note that if A wo generates an analytic semigroup for wo then the Lipschitz propert
48. ravelling wave solutions pulses fronts and wavetrains of quasi linear parabolic reaction diffusion systems of the form 1 1 u a u Ua e f u us t gt 0 ER with unknown u t x R The nonlinearities a f are smooth and a u R is strongly elliptic in the domain of interest but does not have to be symmetric We further consider a variant of 1 1 in higher space dimensions x R up to n 3 The nonlinearities may also depend explicitly on x in an appropriate way Quasi linear reaction diffusion systems arise as models in various contexts due to nonlinear fluxes density dependent diffusion and self or cross diffusion see e g 2 54 55 For pattern formation problems it is natural to consider an extended domain and to neglect the influence of boundary conditions Travelling waves i e solutions of 1 1 constant in a co moving frame x ct with speed c R having constant or periodic asymptotic states are among Received by the editors June 16 2013 accepted for publication in revised form by K Promislow October 22 2013 published electronically February 25 2014 The second and third authors acknowledge support by the Complexity program of the Dutch research fund NWO http www siam org journals siads 13 1 92563 html Martin Luther Universitat Halle Wittenberg Institut fiir Mathematik 06099 Halle Germany martin meyries mathematik uni halle de tUniversitat Bremen Fachbereich Mathemat
49. rbation of which leaves Fredholm properties invariant see 30 Theorem IV 5 26 The point spectrum of a pulse or a front is determined by detecting intersections of the stable and unstable subspaces of v A A v Here the Evans function 1 23 is a powerful tool and we refer readers to the survey 28 59 and the references therein For a wavetrain i e when T is periodic with wavelength period L gt 0 the coefficients of are periodic The point spectrum is empty Instead of the Fourier transform here the Floquet Bloch transform applies and yields see 43 Theorem A 4 also for higher space dimensions 3 3 spec L U spec B k KE 0 2n L For 0 27 L the operator B x H 0 L C Le per 0 L Le per 0 L with periodic boundary conditions is given by K U e Lle U Lik O2 U where C is the formal operator symbol of Since spec B k consists only of eigenvalues its spectrum is fully determined by the solvability of the family of boundary value problems Elin 0 U U U 0 U L 262 M MEYRIES J D M RADEMACHER AND E SIERO In fact also multiplicity of eigenvalues is determined via Jordan chains as in 1 59 Notably the spectrum again comes in curves now an infinite countable union since the eigenvalue problem for each still concerns an unbounded operator rather than a matrix as in the case of a homogeneous steady state Via V e U the boundary valu
50. rem 5 1 5 we write n 1 4 5 T us N ug M us A us X MPT TET us MT us k 0 Denote the right hand side of this by Gn Hn We claim that 4 6 Gall lt a r 2a sin On r Hall lt Cuar where Cm ahs is independent of n To see this we use 4 4 to obtain a n n n n n 4 7 Gal lt 7 M u Re A u Im A of A vl 0m ull IA Q n n Ni oat oy Re M X u iv 2r sin On lt a r N 2a sin On r For the sum H we use 4 2 that T us lt 5ar lt 5a for k lt n 1 and that r n lt r7 to obtain n 1 Hall lt So K r n 8 5ark k 0 1 3 n k 1 a57 K Bron T a lt Cuar re k 0 This shows the claim 4 6 QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 265 Now it follows from 4 5 4 6 and o gt 1 that T us lt 5ar provided that a lt 1 is such that Cha lt 1 By induction for all n 0 N we obtain that T us is well defined and the estimates T u lt 5ar and 4 6 hold true Step 3 As a consequence for dist T us we have to consider only U such that I 2b C lt 10a Indeed for w gt 10a we have T us C gt 5a but T us w 0 T us lt 5a There is some small 7o gt 0 such that 4 8 WC W O C 00 Ie lt 70 where p lt C for a constant C indepen
51. results can be obtained from the unweighted case by a simple similarity transformation Concerning weights we also mention that the approach of 38 has proven useful for quasi linear parabolic problems in weighted spaces see 8 25 68 Invariant manifolds for quasi linear parabolic systems with nonlinear boundary conditions on bounded or exterior domains are constructed e g in 36 37 64 see also the references given there Besides the above approaches based on maximal L and Holder regularity there is a similar abstract approach based on continuous regularity 7 15 Completely different frameworks for problems in weaker settings on bounded domains with boundary conditions are presented in 2 24 They should also be applicable to problems on R Finally the pioneering work of 35 should be mentioned For a comprehensive overview of possible settings for quasi linear parabolic problems we refer the reader to 4 3 Stability and spectra of travelling waves While travelling waves also occur in higher space dimensions we restrict here to x R Throughout let u t U x ct be a travelling wave solution of ut a u ue e f u ux ceER with speed c R and profile 7 BC R RY solving the ODE 2 3 We assume that a f are C and that a is uniformly positively definite in a vicinity of the image of Suitable finite regularity of u a f suffices for each of the following results and we assume infinite smoothnes
52. roach of 14 33 49 based on maximal L regularity but we also highlight the approach of 40 based on maximal H lder regularity Besides reaction diffusion problems the approach of 14 33 49 and its extensions apply successfully to the local theory of free boundary problems and to general parabolic problems with nonlinear boundary conditions Here the geometric theory is well developed and still advances especially for the needs in the context of free boundary problems The approach of 40 also applies to fully nonlinear problems Recently in 50 51 the principle of linearized orbital stability with asymptotic phase for manifolds of equilibria has been established in the quasi linear case for any sufficiently strong well posedness setting see e g 26 section 5 1 for the semilinear case It in particular applies to the orbital stability of pulses and fronts for 1 1 in both approaches mentioned before The conclusion from arbitrary unstable spectrum to nonlinear orbital instability of a manifold of equilibria does not seem to exist in the literature Refining arguments from 26 Theorem 5 1 5 and 61 for single equilibria we close this gap in the present paper This might be of interest also in other contexts where families of equilibria occur In more detail our considerations may be summarized as follows e In one space dimension x R a possible phase space for the evolution under 1 1 of localized perturbations from travell
53. s only for the sake of a simple exposition We further assume that is constant or periodic at infinity and that the asymptotic states are approached exponentially A travelling wave is called a pulse or a front if the asymptotic states are equal or different homogeneous equilibria respectively A wavetrain is a periodic travelling wave and we refer to travelling waves with at least one periodic asymptotic state as generalized fronts or pulses 3 1 Stability in a perturbative setting The evolution of perturbations u of u is gov erned by 3 1 Ut a T U Ug e a U U Ue e C Ue Ux f T u Be Ux QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 259 where the co moving frame x ct is again denoted by x By translation invariance of the underlying equation stability must be considered with respect to the family of translates S u 7 7 ER Theorems 2 4 2 5 and 2 7 guarantee local well posedness of 3 1 for initial data from H X p or X BUC sufficiently close to S Note that in Theorem 2 5 it is actually assumed that f is independent of uz Even though H C Ba P we distinguish between these cases because of the different corresponding base spaces H and L and to highlight that a pure Sobolev space setting suffices for 3 1 For BUC or in case of a pulse one could equivalently consider 3 1 with replaced by zero in a neighborhood of u 7 7 R If uy is a pulse
54. stability of viscous shock waves Indiana Univ Math J 47 1998 pp 741 872
55. t a does not have to be symmetric and that a f may be less regular than actually stated Theorem 2 2 Assume that a RN RN is Ct such that a C RN is positive definite for each C RN and that f RN x RY gt RY is C with f 0 0 0 Then 2 2 is locally well posed in the phase space X H The solutions belong to H J H R N L J H R N C J H R on time intervals J 0 T away from the maximal existence time Proof We choose Xp Ht X H and p 2 Then H H 1 2 9 H see 66 Remark 2 4 2 2 Define the superposition Nemytskii operators A and F by A u v a u vz z and F u f u uz Then F H H and A H gt Z H H are Lipschitz on bounded sets by Lemma A 1 For the generator property let w H be arbitrary Denote by Azz the realization of A w on L with domain H Since wo a wo BC by Sobolev s embedding Ht C BC it follows from 6 Corollary 9 5 that the operator Az2 generates an analytic Co semigroup on L Next let Agi be the realization of A wo on H i e the restriction of Azz to H Since Ht L H 1 95 see again 66 it follows from 39 Theorem 5 2 1 that Aji with domain D Ap u H Azzu H generates an analytic Co semigroup as well Using the algebra property of H it is elementary to check that D Ayi H see the proof of Lemma A 3 in the appendix Thus Theorem 2 1 applies Remark 2 3 Employing e g Angenent s paramet
56. t space 2 2 Well posedness based on maximal H lder regularity We formulate the well posedness result of 40 Chapter 8 for abstract quasi linear parabolic problems 2 9 Ou A uju F u t gt 0 u 0 uo The approach of 40 is based on maximal H lder regularity see also 3 Chapter II 2 for the general linear theory It also covers fully nonlinear problems and does not take into account the quasi linear structure of 2 9 It has the big advantage of being applicable in arbitrary Banach spaces Xo while in applications maximal L regularity is usually restricted to reflexive Banach spaces excluding spaces of continuous functions Moreover the phase space equals the domain of the linearized operator which is usually easier to describe than an interpolation space The following well posedness result for 2 9 is a consequence of 40 Theorem 8 1 1 Propo sition 8 2 3 Corollary 8 3 3 Theorem 2 6 Let Xo X be arbitrary Banach spaces such that X is continuously and densely embedded in Xo Let V C X X be open define F u A u u F u and suppose that e FEC V Xo with locally Lipschitz derivative e for each wo E V the operator F wo with domain X generates a strongly continuous analytic semigroup on Xo and ul x F wo ullx defines an equivalent norm on X1 Then 2 9 is locally well posed in V and solutions are classical in time QUASI LINEAR PARABOLIC REACTION DIFFUSION SYSTEMS 257 As already menti
57. w gt 0 in which 1 2 supports a large family of travelling waves see 65 and section 5 This paper is organized as follows In section 2 different well posedness setting results for 1 1 are treated section 3 is devoted to the spectrum of the linearization in travelling waves The connection to nonlinear stability and instability is considered in section 4 In section 5 we expand the discussion of 1 2 and illustrate the application of the general results For the sake of self containedness we prove some technical results in the appendix Notation All Banach spaces are real and we consider complexifications if necessary We write X 1 Xo for the bounded linear operators between Banach spaces Xo X1 and L Xo L Xo Xo The usual Sobolev spaces based on L R are denoted by H and H H By BOC BC R and BUC BUC R we denote the Banach space of bounded C functions and of bounded C functions such that all derivatives up to order k are uniformly continuous respectively 2 Frameworks for well posedness We formulate the abstract well posedness results based on maximal regularity and present three concrete frameworks for quasi linear reaction diffusion systems In one space dimension we obtain well posedness in H and in BUC and in space dimensions less than or equal to three we have well posedness in certain Besov spaces More general problems and further settings are briefly discussed at the end of this section
58. y of A as in the theorem combined with well known perturbation results for semigroups see 40 Proposition 2 4 2 imply that this is true for any A t with wo in a small neighborhood of wo This gives a candidate for V To verify the assumptions in 33 section 2 and 49 Theorem 3 1 and prove Theorem 2 1 we need only to know that A wo has for each wo V the property of maximal L regularity on finite time intervals J But in Hilbert spaces this already follows from the assumed generator property of A wo Indeed by 20 Theorems 3 3 7 1 it suffices to consider the case p 2 J R4 and that the semigroup generated by A wo is exponentially decaying In this situation maximal L regularity follows from 63 see also 49 Theorem 1 6 for the short proof using Plancherel s theorem 2 1 1 One space dimension Well posedness in H For u t x R we apply the abstract result Theorem 2 1 to the reaction diffusion system 2 2 ut a u uz e f u Us t gt 0 TER 254 M MEYRIES J D M RADEMACHER AND E SIERO To obtain a simple setting with familiar function spaces which is at the same time directly linked to L spectral theory we work with Xo Ht H R as a base space In one space dimension and only there this is possible since Ht is an algebra i e uw H and ue 72 lt Cllullz lollqo for u w Ht We start with the case when the nonlinearities in 2 2 are everywhere defined We em phasize tha
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