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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.

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