On Global Solutions and Blow-Up of Solutions for a Nonlinearly Damped Petrovsky System

Abstract

We consider the initial boundary value problem for a Petrovsky system with nonlinear dampingin a bounded domain. We showed that the solution is global in time under some conditions without the relation between m and p. We also prove that the local solution blows-up in finite time if p > m and the initial energy is nonngeative. The decay estimates of the energy function and the estimates of the lifespan of solutions are given. In this way, we can extend the result of ([6]).

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.…”

“…The proof of global existence result is based on the potential well theory and the continuous principle; while for energy decay result, the proof is based on the Nakao's inequality and some techniques given in [43].…”

Blow-Up of Solutions for a System of Petrovsky Equations with an Indirect Linear Damping

“…where a, b are defined as (12) and (13), respectively. For the case (iv), by the steps in case (i), we get (37) and (38) if and only if…”

“…In [10], Messaoudi studied problem (3) with g(u t ) = α |u t | p−1 u t and showed that the solution blows up in finite time if r > p and the energy is negative, while the solution is global if p ≥ r. Then Wu and Tsai in [12] showed that the solution is global under some conditions without any relation between p and r. They also proved the local solution blows up in finite time if r > p and the initial energy is nonnegative. In [9], Amroun and Benaissa proved the global existence of the solutions by means of the stable set method in H 2 0 (Ω ) combined with the Faedo-Galerkin procedure.…”

Blow up of the solutions for the Petrovsky equation with fractional damping terms

“…Wu and Tsai [11] obtained global existence and blow up of the solution of the problem (1.2). Later, Chen and Zhou [2] studied blow up of the solution of the problem (1.2) for positive initial energy.…”