Abstract
In this paper, we mainly use operator decomposition technique to prove the global attractors which in for the Kirchhoff wave equation with strong damping and critical nonlinearities, are also bounded in .
Highlights
L2 in the sense of “partially strong topology”
Wang [12] studied the longtime behavior of the Kirchhoff type equation with a strong dissipation and proved that the continuous semigroup S (t ) possessed global attractors in the phase spaces with low regularity
The purpose of this paper is to prove the global attractor of problem (1.1)-(1.2), which attracts every
Summary
Let φ= (ξ ) f (ξ ) + λξ , problem (1.1)-(1.2) becomes utt. The Young’s inequality with ε : Let a > 0,b > 0,ε > 0, p > 1, q > 1 , and 1 + 1 =1, pq. Throughout this paper, we will denote by C a positive constant which is various in different line or even in the same line and use the following abbreviations:. A subset A of E is called a global attractor for the semigroup, if A is compact and enjoys the following properties: 1) A is invariant, i.e. S (t ) A= A,∀t ≥ 0 ; 2) A attracts all bounded set of E. We only formulate the following results, which is proved in [13]: Lemma 2.3. By exploiting (2.11) and (2.14), we can get u,ut are respectively bounded in
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