×

Generic properties of equilibria of reaction-diffusion equations with variable diffusion. (English) Zbl 0601.35053

Consider the nonlinear scalar parabolic equation with homogeneous Neumann boundary conditions: \[ (P)\quad u_ t=(a^ 2(x)u_ x)_ x+f(u),\quad 0\leq x\leq 1,\quad u_ x(0,t)=u_ x(1,t)=0, \] where \(f: R\to R\) is a \(C^ 2\)-function and \(a^ 2:(0,1)\to R\) is continuous. It is shown that generically for \((a^ 2,f)\) in \(C((0,1),R_+)\times C^ 2(R)\) with the appropriate topology that all the equilibrium solutions are hyperbolic. The structure of the set E of these equilibria is derived by its partition into the sets \[ H=H(a^ 2,f)=\{u\in E:(\frac{\partial u}{\partial u_ 0})(y)\neq 0\text{ for every } y\in (0,1)\quad such\quad that\quad u_ x(y)=0\}, \]
\[ H'=H'(a^ 2,f)=\{u\in E:(\frac{\partial u}{\partial u_ 0})(y)=u_ x(y)=0\text{ for some } y\in (0,1)\}. \] The main results are the following theorems. Theorem A. There exists a residual subset \(G\subset C([0,1],R_+)\times C^ 2_ s(R)\) such that, for each \((a^ 2,f)\in G\), all equilibria of (P) are either hyperbolic or saddle-node degeneracies.
Theorem B. I. For each fixed \(a^ 2\in C([0,1],R_+)\), there exists a residual set \(F(a^ 2)\) in \(C^ 2_ s(R)\) such that, for each \(f\in F(a^ 2)\), all equilibria in \(H(a^ 2,f)\) are either hyperbolic or saddle-node degeneracies.
II. For each fixed \(f\in C^ 2_ s(R)\), there exists a residual set A(f) in \(C([0,1],R_+)\) such that, for each \(a^ 2\in A(f)\), all equilibria in \(H'(a^ 2,f)\) are either hyperbolic or saddle-node degeneracies.
Reviewer: P.McCoy

MSC:

35K57 Reaction-diffusion equations
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abraham, Transversal Mappings and Flows (1967) · Zbl 0171.44404
[2] DOI: 10.1016/0022-0396(84)90172-4 · Zbl 0488.58015
[3] DOI: 10.1016/0022-0396(82)90017-1 · Zbl 0525.34033
[4] Matano, J. Math. Kyoto Univ. 18 pp 163– (1978)
[5] DOI: 10.1016/0022-0396(84)90022-6 · Zbl 0544.34019
[6] Henry, Lecture Notes in Mathematics 840 (1981)
[7] DOI: 10.1016/0362-546X(85)90007-0 · Zbl 0526.35010
[8] Hale, Asymptotic behaviour of gradient-like systems (1982) · Zbl 0542.34027
[9] Coddington, Theory of Ordinary Differential Equations (1955) · Zbl 0064.33002
[10] Peixoto, An. Acad. Brasil Ciênc. 41 pp 1– (1969)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.