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The existence of global attractor for a fourth-order parabolic equation. (English) Zbl 1270.35125

The authors investigate the following fourth-order parabolic equation \[ u_t +\gamma \Delta^2u-\text{div}\;\left(|\nabla u|^{p-2}\nabla u\right)=0, \] where \(\gamma >0, p>2\) and \(\Omega\) is a two-dimensional smooth bounded domain. This equation is supplemented by the boundary conditions \(\partial u/\partial n=0,\;\partial(\Delta u)/\partial n=0\) on \(\partial \Omega\) and the initial condition \(u(x,0)=u_0\). Taking the initial data in any bounded set \(B\) of \[ U_k=\{u:\;u \in H^2(\Omega),\;|m(u)|\leq k\} \] and using the multipliers \(u\), \( \Delta u\) and \(\Delta^2 u\) as well as \(\Delta^2 u\) (tested against \(\Delta\) applied to the equation), the authors show the boundedness of \(L^2\), \(H^1\), \(H^2\) and \(H^3\) norms (respectively) of \(u\) for \(t\) sufficiently large. These estimates imply the existence of a bounded set in \(H^3(\Omega)\) that is a global attractor in \(U_k\). This results is proved for \(7/2 < p \leq 4\). In the remaining of the paper, the authors prove (under same restriction on \(p\)) the existence of a global attractor in \(H_k\) for \(0 \leq k < 5\) attracting any bounded subset of this space. In this paper, the scheme of the proofs and main results are similar to the ones of: X. Zhao and B. Liu [J. Korean Math. Soc. 49, No. 2, 357–378 (2012; Zbl 1242.35058)].

MSC:

35B41 Attractors
35K35 Initial-boundary value problems for higher-order parabolic equations
35K59 Quasilinear parabolic equations

Citations:

Zbl 1242.35058
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References:

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