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

Rocha, Carlos

Proc. R. Soc. Edinb., Sect. A 101, 45-55 (1985).

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

Keywords:

bifurcations; nonlinear scalar parabolic equation; homogeneous Neumann boundary conditions; equilibrium solutions

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)