Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent

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

“…We first give two lemmas. Their proofs are similar to those in [9], and we omit them. Define the energy function…”

“…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.…”

“…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. The essential difference between this paper and [9] is that the origin is not a local minimum point but a saddle point of the corresponding Lyapunov function.…”

“…In [9] the authors have proved the existences of multiple equilibrium points in the global attractor of (1.1) with λ = 0. The essential difference between this paper and [9] is that the origin is not a local minimum point but a saddle point of the corresponding Lyapunov function. Also, compared with the reaction-diffusion equations in [16], the phase space of the wave equation is the product space H 1 0 (Ω) × L 2 (Ω); nevertheless, to apply Theorem 1.1, one key point is to decompose the phrase space, and we present a new strategy to decompose the product space (see the proof of Theorem 3.3 in detail).…”

“…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.…”