Connecting orbits in scalar reaction diffusion equations. II: The complete solution. (English) Zbl 0699.35144

[For part I, see Dyn. Rep. 1, 57-89 (1988; Zbl 0679.35047).]
In part I the authors studied the one-dimensional reaction-diffusion equation \[ u_ t=u_{xx}+f(u),\quad x\in (0,1) \] with Dirichlet boundary conditions \(u(t,0)=u(t,1)=0\), and obtained a partial answer to the question:
(Q) Given a stationary (i.e. time-independent) solution of the above problem, which other stationary solutions does it connect to?
The present paper answers completely this question by reducing the number of exclusion principles to two.
Reviewer: J.Mawhin


35K57 Reaction-diffusion equations


Zbl 0679.35047
Full Text: DOI


[1] Angenent, S, The Morse-Smale property for a semilinear parabolic equation, J. differential equations, 62, 427-442, (1986) · Zbl 0581.58026
[2] Brunovský, P; Chow, S.-N, Generic properties of stationary state solutions of reaction-diffusion equations, J. differential equations, 53, 1-23, (1984) · Zbl 0544.34019
[3] Brunovský, P; Fiedler, B, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear anal., 10, 179-193, (1986) · Zbl 0594.35056
[4] Brunovský, P; Fiedler, B, Connecting orbits in scalar reaction diffusion equations, (), 57-89 · Zbl 0679.35047
[5] Fiedler, B; Brunovský, P, Connections in scalar reaction-diffusion equations with Neumann boundary conditions, (), 123-128 · Zbl 0616.35044
[6] {\scR. Franzosa}, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc., in press. · Zbl 0689.58030
[7] Hale, J; Magalhães, L; Oliva, W, An introduction to infinite dimensional dynamical systems-geometric theory, ()
[8] Henry, D, Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001
[9] Henry, D, Some infinite dimensional Morse-Smale systems defined by parabolic equations, J. differential equations, 59, 165-205, (1985) · Zbl 0572.58012
[10] Palis, J; de Melo, W, Geometric theory of dynamical systems, (1982), Springer-Verlag Berlin/New York
[11] Poláčik, P, Generic hyperbolicity in one-dimensional reaction diffusion equations with general boundary conditions, Nonlinear anal., 11, 593-597, (1987) · Zbl 0639.35039
[12] Smoller, J; Wasserman, A, Generic bifurcation of steady-state solutions, J. differential equations, 52, 432-438, (1984) · Zbl 0488.58015
[13] Smoller, J, Shock waves and reaction-diffusion equations, () · Zbl 0508.35002
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