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Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping. (English) Zbl 1046.37045
The authors consider two types of second-order PDEs in a bounded domain \(\Omega\subset\subset\mathbb R^N\), \(N\geq2\) both of which have a gradient structure (possess a global Lyapunov function). The first is a damped semilinear hyperbolic problem \[ u_{tt}+\alpha u=\Delta u+g(x,u),\;x\in\Omega,\;\;t>0,\;\;u(x,0)=u_0,\;u_t(x,0)=v_0(x), \alpha>0, \tag{1} \] and the second one is an (ill-posed) elliptic problem \[ -u_{tt}+\alpha u_t=\Delta u+g(x,u),\;t>0,\;\;u(x,0)=u_0,\;\;\alpha\neq0, \tag{2} \] endowed by the Dirichlet boundary conditions.
The main result of the paper is the fact that, for every sufficiently smooth bounded domain \(\Omega\) of \(\mathbb R^N\) with \(N\geq2\) there exists an \(C^\infty\)-nonlinearity \(g(x,u)\) such that problems (1) and (2) possess bounded solutions (as \(t\to\infty\)) which do not stabilize to a single equilibrium. The proof of that result is based on the two-dimensional centre manifold reduction and on the special construction of smooth planar gradient vector fields with nonstabilizing trajectories introduced in [P. Polácik and K. P. Rybakowski, J. Differ. Equations 124, 472–494 (1996; Zbl 0845.35054)].

37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35L70 Second-order nonlinear hyperbolic equations
35J60 Nonlinear elliptic equations
35B35 Stability in context of PDEs
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