On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term

Abstract

In this paper we consider a singular nonlocal viscoelastic problem with a nonlinear source term and a possible damping term. We proved that if the initial data enter into the stable set, the solution exists globally and decays to zero with a more general rate, and if the initial data enter into the unstable set, the solution with non-positive initial energy as well as positive initial energy blows up in finite time. These are achieved by using the potential well theory, the modified convexity method and the perturbed energy method.

Global Existence, Asymptotic Behavior and Blow-up of Solutions for a Suspension Bridge Equation with Nonlinear Damping and Source Terms

“…Under some assumptions, he showed the solution of (1.9) decays exponentially if g behaves like a linear function, whereas the decay is polynomially otherwise. For more decay results, we refer the reader to [5,6,7,10,16,19,25,26,40,43,47,48] and the references therein. In recent years, more authors pay attention to the lower and upper bounds for blow-up time.…”

Global Existence, Exponential Decay and Finite Time Blow-up of Solutions for a Class of Semilinear Pseudo-parabolic Equations with Conical Degeneration

“…In this subsection, we establish finite time blow-up of solution for problem (1.1) when J(u 0 ) < d and I(u 0 ) < 0 by using the concavity argument (see [13,14,15]) and properties of a family of potential wells. Furthermore, by making use of a differential inequality technique (see [18]) we determine a lower bound on blow-up time for certain solutions of problem (1.1) if blow-up occurs.…”

Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized damping term

“…The purpose of this paper is to obtain, motivated by the previous mentioned work with Liu et al and Mesloub and Messaoudi, we prove that the same previous study in Draifia et al can be extended to damping terms a ( x ) u t (resp. a ( x ) v t ) using the potential well theory as well as with respect some different assumptions.…”