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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.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations
##### Keywords:
Łojasiewicz-Simon type inequality