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This paper is concerned with the existence of global weak solution for a nonlinear viscoelastic problem with internal damping and a logarithmic source term and Dirichlet boundary initial conditions, and with the study of the asymptotic behavior of the problem, involving: a) exponential decay of total energy of solutions for initial data in the set of stability created by the Nehari manifold, b) the exponential growth of the logarithmic source term for negative initial energy. In the existence of global weak solution we employed similar ideas as in the work of S. Cordeiro, J. Ferreira, et al., 2021, where the Faedo-Galerkin method was combined with Aubin-Lions lemmas for the passage to the limit in the nonlinear terms. In the study of the exponential decay of the total energy and in the growth of the logarithmic term of the energy we adapted the perturbed energy methods in a work of Messaoudi & Tatar, 2006 and 2003. [ABSTRACT FROM AUTHOR]

This paper is concerned with blow up of weak solutions of the following nonlinear viscoelastic problem with internal damping and logarithmic source term |ut|ρutt + M(∥u∥2)(-∆u) - ∆utt + Z0t g(t - s)∆u(s)ds + ut = u|u|p R-2 ln |u|k R with Dirichlet boundary initial conditions in a bounded domain Ω ⊂ Rn. In the physical point of view, this is a type of problems that usually arises in viscoelasticity. It has been considered with power source term first by Dafermos [3], in 1970, where the general decay was discussed. We establish conditions of p, ρ and the relaxation function g, for that the solutions blow up in finite time for positive and nonpositive initial energy. We extend the result in [15] where is considered M = 1 and external force type |u|p-2u in it. Further we state and sketch the proof of a result of local existence of weak solution that is used in the proof of the theorem on blow up. The idea underlying the proof of local existence of solution is based on Faedo-Galerkin method combined with the Banach fixed point method.

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In this paper, we consider a nonlinear boundary value problem for viscoelastic equations with a source term. By basing on Faedo-Galerkin approximations and compactness argument, this work is devoted to prove the existence, uniqueness, and also continuous dependence with respect to the initial data of solutions. [ABSTRACT FROM AUTHOR]

In this article, we investigate a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density. We prove the global existence of weak solutions and general decay of the energy by using the Faedo–Galerkin method [Z.Y. Zhang and X.J. Miao,Global existence and uniform decay for wave equation with dissipative term and boundary damping, Comput. Math. Appl. 59 (2010), pp. 1003–1018; J.Y. Park and J.R. Kang,Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Acta Appl. Math. 110 (2010), pp. 1393–1406] and the perturbed energy method [Zhang and Miao (2010); X.S. Han, and M.X. Wang,Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. TMA. 70 (2009), pp. 3090–3098], respectively. Furthermore, for certain initial data and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. This result is an improvement over the earlier ones in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a Cauchy problem for a nonlinear viscoelastic equation with nonlinear damping and source terms. Under suitable assumptions on the initial data and the relaxation function, we establish a finite-time blow-up result and a global existence result. [Copyright &y& Elsevier]

Abstract: In this paper we investigate a nonlinear viscoelastic equation with linear damping. Global existence of weak solutions and the uniform decay estimates for the energy have been established. [Copyright &y& Elsevier]

Abstract: We consider the following nonlinear viscoelastic equation: in a bounded domain for . We show that the dissipation induced by the integral term is strong enough to stabilize the solution. This result improves an earlier one given by Cavalcanti et al. in [M.M. Cavalcanti, V.N.D. Cavalcanti, J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001) 1043–1053]. [Copyright &y& Elsevier]