Abstract
In this paper, we consider the initial boundary problem for the Kirchhoff type wave equation. We prove that the Kirchhoff wave model is globally well-posed in the sufficiently regular space \begin{document}$ (H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega) $\end{document} , then, we also obtain that the semigroup generated by the equation has a global attractor in the corresponding phase space, in the presence of a quite general nonlinearity of supercritical growth.
Highlights
We consider the following Kirchhoff wave model in a bounded domain Ω ⊂ Rn with smooth boundary ∂Ω:∂ttu − σ( ∇u 2)∆∂tu − φ( ∇u 2)∆u + f (u) = h(x), in Ω × R+, u|∂Ω = 0, u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω. (1.1)Here, h ∈ L2(Ω) is an external force term, f (u) is a given source term, σ and φ are nonlinear scalar functions which will be specified later.One dimensional model (1.1) with φ(s) = a + bs, σ(s) ≡ 0, f (u) ≡ 0, was introduced by Kirchhoff [12] to describe small vibrations of an elastic stretched string
Many mathematicians and physicists paid their attentions in this type of problem, and there have been a lot of results on the nonlinear strongly damped wave equation, see [2, 3, 5, 7, 22, 23, 25] and references therein
We [16] proved the existence of a global attractor for problem (1.1) with non-supercritical nonlinearity (p ≤ p∗), which attracts H01(Ω) × L2(Ω) -bounded set with respect to the H01(Ω) × H01(Ω) norm
Summary
We consider the following Kirchhoff wave model in a bounded domain Ω ⊂ Rn with smooth boundary ∂Ω:. Strong solution, global attractor, semigroup, ωlimit compactness. We [16] proved the existence of a global attractor for problem (1.1) with non-supercritical nonlinearity (p ≤ p∗), which attracts H01(Ω) × L2(Ω) -bounded set with respect to the H01(Ω) × H01(Ω) norm. Under some additional nondegeneracy assumptions (φ > 0), they proved the existence of a finite-dimensional global attractor for problem (1.1) in the natural energy space (H01(Ω) ∩ Lp+1(Ω)) × L2(Ω) endowed with a partially strong topology. In this paper, inspiring by above researches, we first prove the existence and uniqueness of strong solutions to the problem (1.1) in the regular phase space. The proof of existence of the weak solution part is given by a similar argument as in Chueshov [4], where c0|u|p−1 − c1 ≤ f (u) ≤ c2|u|p−1 + c3 with c0 > 0 in the supercritical case.
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