Asymptotic stability of solutions to a semilinear viscoelastic equation with analytic nonlinearity
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Journal of Evolution Equations
Asymptotic stability of solutions to a semilinear viscoelastic equation with analytic nonlinearity Hassan Yassine
Abstract. By a small perturbation of the usual Lyapunov energy and by using the Łojasiewicz–Simon inequality, we show that the dissipation given by the memory term is strong enough to prove the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution to the semilinear viscoelastic equation |u t |ρ u tt − u tt − u +
τ 0
k(s)u(t − s)ds + f (x, u) = g, τ ∈ {t, ∞},
in a bounded regular domain of Rn with Dirichlet boundary conditions. Here, the kernel function k > 0 is assumed to decay exponentially at infinity, the nonlinearity f is analytic in the second variable, and the forcing term g is supposed to decay polynomially or exponentially at infinity. The present work extends the previous result where the given equation was studied in the presence of an additional strong linear damping − u t .
1. Introduction and assumptions In this paper, we study the convergence and decay rate to a steady state of global bounded solutions of the following nonautonomous semilinear viscoelastic equation with finite memory t ⎧ |u t |ρ u tt − u tt − u + 0 k(s)u(t − s)ds + f (x, u) = g in R+ × , ⎪ ⎪ ⎨ u = 0 on R+ × , (1) ⎪ u(0) = u 0 in , ⎪ ⎩ u t (0) = u 1 in , and with infinite memory ⎧ ρ u − u − u + ∞ k(s)u(t − s)ds + f (x, u) = g in R+ × , |u | t tt tt ⎪ 0 ⎪ ⎨ u = 0 on R+ × , (2) ⎪ u(−t) = u 0 (t) for t ≥ 0, ⎪ ⎩ u t (0) = u 1 in . Here, ⊆ Rn (n ≥ 1) is a bounded open set with smooth boundary , ρ is a real 2 if n ≥ 3 and ρ ≥ 0 if n ∈ {1, 2}, and the functions number such that 0 ≤ ρ ≤ n−2
H. Yassine
J. Evol. Equ.
u 0 , u 1 : → R and u 0 : R+ × → R are given initial data. The relaxation function k, the nonlinearity f and the forcing term g will be specified later. This type of problem arises in viscoelasticity and has been widely studied in the literature. Several results concerning existence, uniqueness, convergence to equilibrium, and blow-up of solutions have recently been established. For the autonomous linear case (i.e. f = g = 0), the first result appears in [5], where a strong linear damping (−u t ) is added; the global existence and exponential decay of weak solutions are established assuming an exponentially decaying memory kernel k. Later, it was shown in [10,18] that the strong damping could be replaced by a linear or nonlinear weak damping (u t or |u t |m u t ). In this direction, we refer the reader also to [17], the recent abstract result given by Cavalcanti et al. [6], and the references therein. For a monotone nonlinearity like f (u) = |u| p u, a number of papers have appeared covering a large class of kernels k, which guarantee stability and show the relation between the decay rate of k and the asymptotic behaviour of solutions of the considered problem. Examples of such results can be found in [3,15] where the nonlinearity f is competing with the dissipation terms induced by both the viscoel
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