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Some oscillatory properties of solutions to second order evolution equations. (English) Zbl 0816.35087

Lumer, Günter (ed.) et al., Partial differential equations. Models in physics and biology. Contributions to the conference, held in Han-sur- Lesse, Belgium, in December 1993. Berlin: Akademie Verlag. Math. Res. 82, 159-165 (1994).
Summary: The main objective of this survey paper is to record some results and open problems concerning the asymptotic behaviour of solutions to \[ u_{tt}- \Delta u+g(u) =0 \quad \text{in } \mathbb{R}\times \Omega, \qquad u=0 \quad \text{on }\mathbb{R} \times \partial\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), \(g\in C^ 1 (\mathbb{R})\) and \(g\) satisfies the sign condition \(\forall u\in \mathbb{R}\), \(g(u) u\geq 0\). As a motivation, we first consider an abstract version of the wave equation (case \(g=0\)).
For the entire collection see [Zbl 0809.00019].

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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