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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

${u}_{t}={\Delta }u+f\left(x,u\right),\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{4pt}{0ex}}x\in {\Omega },\phantom{\rule{1.em}{0ex}}u=0,\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{4pt}{0ex}}x\in \partial {\Omega }·\phantom{\rule{2.em}{0ex}}\left(1\right)$

Here ${\Omega }$ is a bounded domain in ${ℝ}^{N}$ with smooth boundary and $f$ is a sufficiently regular function on $\overline{{\Omega }}×ℝ$. Problem (1) defines a local semiflow on an appropriate Banach space, for example, the Sobolev space ${W}_{0}^{1,p}\left({\Omega }\right)$ with $p>N$. This semiflow is gradient-like: the energy functional

$\varphi ↦{\int }_{{\Omega }}\left(\frac{1}{2}{|\nabla \varphi \left(x\right)|}^{2}-F\left(x,\varphi \left(x\right)\right)\right)dx,$

where $F\left(x,u\right)$ is the antiderivative of $f\left(x,u\right)$ 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 $𝔊$ denote the space of all ${C}^{k}$ functions $f:\overline{{\Omega }}×ℝ\to ℝ$ endowed with the ${C}^{k}$ Whitney topology. This is the topology in which the collection of all the sets

$\left\{g\in 𝔊:|{D}^{i}f\left(x,u\right)-{D}^{i}g\left(x,u\right)|<\delta \left(u\right),\phantom{\rule{4pt}{0ex}}i=0,\cdots ,k,\phantom{\rule{4pt}{0ex}}x\in \overline{{\Omega }},\phantom{\rule{4pt}{0ex}}u\in ℝ\right\},$

where $\delta$ is a positive continuous function on $ℝ$, forms a neighborhood basis of an element $f$. Recall that $𝔊$ is a Baire space: any residual set is dens in $𝔊$. Our main result reads as follows.

Theorem. There is a residual set ${𝔊}^{\text{MS}}$ in $𝔊$ such that for any $f\in {𝔊}^{\text{MS}}$ all equilibria of (1) are hyperbolic and if ${\varphi }^{-}$, ${\varphi }^{+}$ are any two such equilibria then their stable and unstable manifolds intersect transversally.

MSC:
 35K65 Parabolic equations of degenerate type 37D15 Morse-Smale systems 35K57 Reaction-diffusion equations 35-99 Partial differential equations (PDE) (MSC2000)