Existence of multiple equilibrium points in global attractor for damped wave equation. (English) Zbl 1503.35113
Summary: This paper is a continuation of F. Meng and C. Zhong in [Discrete Contin. Dyn. Syst., Ser. B 19, No. 1, 217–230 (2014; Zbl 1288.35101)]. We go on studying the property of the global attractor for some damped wave equation with critical exponent. The difference between this paper and [loc. cit.] is that the origin is not a local minimum point but rather a saddle point of the Lyapunov function \(F\) for the symmetric dynamical systems. Using the abstract result established in [J. Zhang et al., Nonlinear Anal., Real World Appl. 36, 44–55 (2017; Zbl 1364.35048)], we prove the existence of multiple equilibrium points in the global attractor for some wave equations under some suitable assumptions in the case that the origin is an unstable equilibrium point.
MSC:
| 35L71 | Second-order semilinear hyperbolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B41 | Attractors |
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
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