The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension. (English) Zbl 0868.35062

Consider the Dirichlet problem for the reaction-diffusion equation \[ u_t= \Delta u+f(x,u),\quad t>0,\;x\in\Omega,\quad u=0,\quad t>0,\;x\in\partial\Omega.\tag{1} \] Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary and \(f\) is a sufficiently regular function on \(\overline\Omega\times\mathbb{R}\). Problem (1) defines a local semiflow on an appropriate Banach space, for example, the Sobolev space \(W^{1,p}_0(\Omega)\) with \(p>N\). This semiflow is gradient-like: the energy functional \[ \varphi\mapsto \int_\Omega(\textstyle{{1\over 2}}|\nabla \varphi(x)|^2- F(x,\varphi(x)))dx, \] where \(F(x,u)\) is the antiderivative of \(f(x,u)\) with respect to \(u\), decreases along nonconstant trajectories. In higher space dimensions, stable and unstable manifolds of hyperbolic equilibria can intersect nontransversally. One of the main objectives of the present paper is to prove that generically this cannot happen. To formulate the result precisely, let \(k\) be a positive integer and let \(\mathfrak G\) denote the space of all \(C^k\) functions \(f:\overline\Omega\times\mathbb{R}\to\mathbb{R}\) endowed with the \(C^k\) Whitney topology. This is the topology in which the collection of all the sets \[ \{g\in{\mathfrak G}:|D^if(x,u)- D^ig(x,u)|< \delta(u),\;i=0,\dots,k,\;x\in\overline\Omega,\;u\in\mathbb{R}\}, \] where \(\delta\) is a positive continuous function on \(\mathbb{R}\), forms a neighborhood basis of an element \(f\). Recall that \(\mathfrak G\) is a Baire space: any residual set is dens in \(\mathfrak G\). Our main result reads as follows.
Theorem. There is a residual set \({\mathfrak G}^{\text{MS}}\) in \(\mathfrak G\) such that for any \(f\in{\mathfrak G}^{\text{MS}}\) all equilibria of (1) are hyperbolic and if \(\varphi^-\), \(\varphi^+\) are any two such equilibria then their stable and unstable manifolds intersect transversally.


35K65 Degenerate parabolic equations
37D15 Morse-Smale systems
35K57 Reaction-diffusion equations
Full Text: DOI


[1] Abraham, R.; Marsden, J. E.; Ratiu, T., Manifolds, Tensor Analysis and Applications (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0875.58002
[2] Abraham, R.; Robbin, J., Transversal Mappings and Flows (1967), Benjamin: Benjamin New York · Zbl 0171.44404
[3] Alessandrini, G.; Vessella, S., Local behavior of solutions of parabolic equations, Comm. Partial Differential Equations, 13, 1041-1057 (1988) · Zbl 0649.35040
[4] Amann, H., Linear and Quasilinear Parabolic Problems (1995), Birkhäuser: Birkhäuser Berlin
[5] Angenent, S. B., The Morse-Smale property for a semilinear parabolic equation, J. Differential Equations, 62, 427-442 (1986) · Zbl 0581.58026
[6] Babin, A. V.; Vishik, M. I., Regular attractors of semigroups of evolutionary equations, J. Math. Pures Appl., 62, 441-491 (1983) · Zbl 0565.47045
[7] Bates, P. W.; Lu, K., A Hartman-Grobman theorem for Cahn Hilliard equations and phase-field equations, J. Dynam. Differential Equations, 6, 101-145 (1994) · Zbl 0795.35052
[8] Blazquez, C. M., Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal., 10, 1277-1291 (1986) · Zbl 0612.58040
[9] Brunovský, P.; Chow, S.-N., Generic properties of stationary solutions of reaction diffusion equations, J. Differential Equations, 53, 1-23 (1984) · Zbl 0544.34019
[12] Chen, M.; Chen, X.-Y.; Hale, J. K., Structural stability for time-periodic one-dimensional parabolic equations, J. Differential Equations, 96, 355-418 (1992) · Zbl 0779.35061
[13] Chow, S.-N.; Lu, K., Smooth invariant foliations in infinite dimensional spaces, J. Differential Equations, 94, 266-291 (1991) · Zbl 0749.58043
[14] Coppel, W. A., Dichotomies in Stability Theory (1978), Springer-Verlag: Springer-Verlag Berlin · Zbl 0376.34001
[15] Daners, D.; Koch Medina, P., Abstract Evolution Equations, Periodic Problems and Applications (1992), Longman: Longman Harlow · Zbl 0789.35001
[16] Eirola, T.; Pilyugin, S. Y., Pseudotrajectories generated by a discretization of a parabolic equation, J. Dynam. Differential Equations, 8, 281-297 (1996) · Zbl 0857.34046
[17] Engelking, R., General Topology (1985), PWN: PWN Warsaw
[19] Golubitsky, M.; Guillemin, V., Stable Mappings and Their Singularities (1974), Springer-Verlag: Springer-Verlag New York · Zbl 0294.58004
[20] Hale, J. K.; Lin, X.-B., Heteroclinic orbits for retarded functional differential equations, J. Differential Equations, 65, 175-202 (1986) · Zbl 0611.34074
[21] Hale, J. K.; Magalhães, L. T.; Oliva, W. M., An Introduction to Infinite-Dimensional Dynamical Systems—Geometric Theorey (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0533.58001
[22] Hale, J. K.; Raugel, G., Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71, 33-95 (1992) · Zbl 0840.35044
[23] Han, Q.; Lin, F. H., Nodal sets of solutions of parabolic equations, II, Comm. Pure Appl. Math., 47, 1219-1238 (1994) · Zbl 0807.35052
[24] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0456.35001
[25] Henry, D., Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 59, 165-205 (1985) · Zbl 0572.58012
[27] Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag Berlin · Zbl 0148.12601
[28] Koch, H., Finite dimensional aspects of semilinear parabolic equations, J. Dynam. Differential Equations, 8, 177-202 (1996) · Zbl 0860.35056
[29] Kupka, I.; Oliva, W. M., Dissipative mechanical systems, Resenhas, 1, 69-115 (1993) · Zbl 0852.58004
[30] Lin, X.-B., Exponential dichotomies and homoclinic orbits in functional differential equations, J. Differential Equations, 63, 227-254 (1986) · Zbl 0589.34055
[32] Lu, K., A Hartman-Grobman theorem for reaction-diffusion equations, J. Differential Equations, 93, 364-394 (1991) · Zbl 0767.35039
[34] Lunardi, A., Analyticity of the maximal solution of an abstract nonlinear parabolic equation, Nonlinear Anal., 6, 505-521 (1982) · Zbl 0486.35017
[35] Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems (1995), Birkhäuser: Birkhäuser Berlin · Zbl 0816.35001
[36] Miranda, C., Partial Differential Equations of Elliptic Type (1970), Springer-Verlag: Springer-Verlag Berlin · Zbl 0198.14101
[37] Mora, X.; Solà-Morales, J., The singular limit dynamics of semilinear wave equations, J. Differential Equations, 78, 261-307 (1989) · Zbl 0699.35177
[39] Palis, J., On Morse-Smale dynamical systems, Topology, 8, 385-404 (1989) · Zbl 0189.23902
[40] Palis, J.; de Melo, W., Geometric Theory of Dynamical Systems (1982), Springer-Verlag: Springer-Verlag New York
[41] Palis, J.; Smale, S., Structural stability theorems, Global Analysis. Global Analysis, Proceedings of Symposic in Pure Mathematics, Vol. 14 (1970), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0214.50702
[42] Palmer, K., Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55, 225-256 (1984) · Zbl 0508.58035
[43] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023
[45] Poláčik, P., Generic hyperbolicity in one-dimensional reaction diffusion equations with general boundary conditions, Nonlinear Anal., 11, 593-597 (1987) · Zbl 0639.35039
[46] Poláčik, P., Transversal and nontransversal intersections of stable and unstable manifolds in reaction diffusion equations on symmetric domains, Differential Integral Equations, 7, 1527-1545 (1994) · Zbl 0809.35041
[49] Quinn, F., Transversal approximation on Banach manifolds, Proceedings of Symposic in Pure and Applied Mathematics (1970), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0206.25705
[50] Quittner, P., Singular sets and number of solutions of nonlinear boundary value problem, Commen. Math. Univ. Carolin., 24, 371-385 (1983) · Zbl 0529.35027
[52] Rocha, C., Generic properties of equilibria of reaction-diffusion equations with variable diffusion, Proc. Roy. Soc. Edinburgh Ser. A, 101, 45-55 (1985) · Zbl 0601.35053
[53] Rynne, B. P., Genericity of hyperbolicity and saddle-node bifurcations in reaction-diffusion equations depending on a parameter, J. Appl. Math. Phys. (ZAMP), 47, 730-739 (1996) · Zbl 0861.35010
[54] Salamon, D., Morse theory, the Conley index and Floer homology, Bull. London Math. Soc., 22, 113-140 (1990) · Zbl 0709.58011
[55] Saut, J.; Temam, R., Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4, 293-319 (1979) · Zbl 0462.35016
[56] Schwarz, M., Morse Homology (1993), Birkhäuser: Birkhäuser Basel/Boston
[57] Shub, M., Global Stability of Dynamical Systems (1987), Springer-Verlag: Springer-Verlag New York
[58] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1967), Springer-Verlag: Springer-Verlag New York
[59] Takens, F., Mechanical and gradient systems: Local perturbation and generic properties, Bol. Soc. Bras. Mat., 14, 147-162 (1983) · Zbl 0572.58008
[60] Uhlenbeck, K., Generic properties of eigenfunctions, Amer. J. Math., 98, 1059-1078 (1976) · Zbl 0355.58017
[61] Zhang, W., The Fredholm alternative and exponential dichotomies for parabolic equations, J. Math. Anal. Appl., 191, 180-201 (1995) · Zbl 0832.34050
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