Existence of multiple equilibrium points in global attractor for damped wave equation

Abstract

This paper is a continuation of Meng and Zhong in (Discrete Contin. Dyn. Syst., Ser. B 19:217-230, 2014). We go on studying the property of the global attractor for some damped wave equation with critical exponent. The difference between this paper and Meng and Zhong in (Discrete Contin. Dyn. Syst., Ser. B 19:217-230, 2014) 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 Zhang et al. in (Nonlinear Anal., Real World Appl. 36:44-55, 2017), 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: 35L05; 37L05; 35B40; 35B41; 58J20

Multiple positive solutions for biharmonic equation of Kirchhoff type involving concave-convex nonlinearities

“…Recently, the authors in [19,21,33,35] proved the existence of the multiple equilibrium points in the global attractors for the symmetric dynamical systems by estimating the lower bound of Z 2 index of two disjoint subsets of the global attractor for which one subset is located in the area where the Lyapunov function F is positive and the other subset is located in the area where the Lyapunov function F is negative. By the way, a fixed point, or a stationary point, or an equilibrium point for a semigroup of an evolutionary equation corresponds to the solution of the related stationary equation [24].…”