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Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains. (English) Zbl 1024.35046
The authors study the following heat equation: $u_t=\Delta u+g(x,u),\quad x\in\Omega,\quad u\big|_{\partial\Omega}=0 \tag{1}$ where $$\Omega$$ a bounded $$C^2$$-smooth domain of $$\mathbb R^N$$.
We recall that equation (1) possesses a global Lyapunov function and, consequently, the $$\omega$$-limit set of every bounded solution of (1) consists only of equilibria. Nevertheless, for every domain $$\Omega$$, the authors find a $$C^\infty$$-function $$g(x,u)$$ such that equation (1) possesses nonconvergent (as $$t\to\infty$$) bounded solutions. Moreover, for the nonlinearity constructed, equation (1) possesses an infinite-dimensional manifold of nonconvergent solutions.

##### MSC:
 35K57 Reaction-diffusion equations 37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
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##### References:
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