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Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition. (English) Zbl 1145.35406
The author deals with the following semilinear parabolic equation \[ u_t-\Delta u+ f(u)= 0,\quad (x,t)\in \Omega\times \mathbb{R}^+ \] subject to the dynamical boundary condition \[ \partial_v u+\mu u+{\partial u\over\partial t}= 0,\quad (x,t)\in \Gamma\times \mathbb{R}^+ \] and the initial condition \[ u|_{t=0}= u_0(x),\quad x\in\Omega, \] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary \(\Gamma\), \(\nu\) is the outward normal direction to the boundary and \(\mu\in \{0,1\}\); and \(f\) is analytic function. The goal of the author is to prove the convergence of a global solution to an equilibrium as \(t\to\infty\). To this end the author uses a suitable Łojasiewicz-Simon type inequality.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations