The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems II

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

“…As an application of Theorems 1.1 and 1.2, the authors have considered the reactiondiffusion equations in [16] and the p-Laplacian equation in [7].…”

“…The current paper is mostly related to [9] and motivated by [16]. In [9] the authors have proved the existences of multiple equilibrium points in the global attractor of (1.1) with λ = 0.…”

“…To the best of our knowledge, there are few related results. Fortunately, recently, the authors in [18] and [16] established the criterions to show the existence of the multiplicity of equilibrium points under some proper conditions.…”

“…In detail, if the semigroup {S(t)} t≥0 is odd and the Lyapunov function F is even, and if the origin is a strictly local minimum point of F, then the authors in [18] have 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. As applications, the authors have considered the reaction-diffusion equations [18], wave equations [8,9], and p-Laplacian equation [15] and proved that the corresponding semigroups {S(t)} t≥0 possess at least 2n pairs of different fixed points in the global attractor A.…”

“…On the other hand, if the origin is no longer a strictly local minimum point but a saddle point of F, then we cannot estimate the lower bound of Z 2 index of the global attractor by the method or technique in [18]. Fortunately, in [16], by proving a new lemma, which is analogous to the intersection lemma in [13], we can still prove the existence of multiple fixed points in a global attractor for some symmetric semigroups with a Lyapunov function F. To be precise, in [16] the authors have mainly obtained the following results.…”