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.
Highlights
In this paper, we consider the following weakly damped wave equation:⎧ ⎪⎪⎨utt + ut – u – λu + φ(u) = 0, (x, t) ∈ Ω × R+,⎪⎪⎩uu(=x,00,) = u0(x), ut(x, 0) = u1(x)(x, t) ∈ ∂Ω × R+, x ∈ Ω, (1.1)where Ω ⊂ R3 is a bounded domain with smooth boundary ∂Ω
We study the properties of the global attractor and obtain the existence of multiple equilibrium points in the global attractor in H01(Ω) × L2(Ω)
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
Summary
We consider the following weakly damped wave equation:. be the sequence of eigenvalues of – on H01(Ω), and let ej be the eigenfunctions corresponding to λj, j = 1, 2, . . . . Here we assume that λ ∈ (λm, λm+1). As the phase spaces we consider are usually Hilbert or Banach spaces, the global attractor is connected In this case, each complete bounded orbit θ (t) is always connected to some pair of fixed points of a semigroup {S(t)}t≥0, and θ (t) is contained in the unstable manifold from the one fixed point (see Def. 2.6) and the stable manifold (see Def. 2.7) from the other fixed point. 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. Under the assumptions of Theorem 1.1, the semigroup {S(t)}t≥0 possesses at least dim X– – codim X+ pairs of different fixed points in A ∩ F–1((0, ∞)).
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