Abstract
In this paper the long time behaviour of the solutions of the 3-Dstrongly damped wave equation is studied. It is shown that thesemigroup generated by this equation possesses a global attractorin $H_{0}^{1}(\Omega )\times L_{2}(\Omega )$ and then it is provedthat this is also a global attractor in $(H^{2}(\Omega )\capH_{0}^{1}(\Omega ))\times H_{0}^{1}(\Omega )$.
Highlights
We consider the following initial-boundary value problem for the strongly damped wave equation: wtt − ∆wt + σ(w)wt − ∆w + f (w) = g(x) in (0, ∞) × Ω, (1.1)w=0 on (0, ∞) × ∂Ω, (1.2)w(0, ·) = w0, wt(0, ·) = w1 in Ω, (1.3)where Ω ⊂ R3 is a bounded domain with sufficiently smooth boundary and g ∈ L2(Ω).As shown in [6] and [13], equation (1.1) is related to the following reactiondiffusion equation with memory:t wt(t, x) = K(t, s)∆w(s, x)ds − f (w(t, x)) + g(x). (1.4)
In [8], the authors investigated the weak attractor for the quasi-linear strongly damped equation wtt − ∆wt − ∆w + f (w) = ∇ · φ′(∇w) + g under the following conditions on the nonlinear functions f and φ: f ∈ C1(R), − C + a1 |s|q ≤ f ′(s) ≤ C |s|q, ∀s ∈ R, φ ∈ C2(R3, R), a2
In this paper we prove the existence of the global attractors for (1.1)-(1.3)
Summary
In [5] and [19], the regularity of the global attractor of (1.5) was proved under the following weaker condition on the source term: f. In [8], the authors investigated the weak attractor for the quasi-linear strongly damped equation wtt − ∆wt − ∆w + f (w) = ∇ · φ′(∇w) + g under the following conditions on the nonlinear functions f and φ: f ∈ C1(R), − C + a1 |s|q ≤ f ′(s) ≤ C |s|q , ∀s ∈ R, φ ∈ C2(R3, R), a2. In [3], the authors have studied the global attractor for the strongly damped abstract equation wtt + D(w, wt) + Aw + F (w) = 0.
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