Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation
Tóm tắt
We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation:
$$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$
in
$${\mathbb {R}}^+\times \Omega $$
, with Dirichlet boundary conditions, where
$$\Omega $$
is a bounded domain in
$${\mathbb {R}}^n$$
and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the Łojasiewicz–Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the term g.
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