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
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