Abstract
Abstract The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E(0) < d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) > 0 (ω = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, infinite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy infinite time blow up result is established.
Highlights
Introduction and main resultsIn this paper, we study initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic source term utt − ∆u − ω∆ut + μut = u ln |u|, (x, t) ∈ Ω × (, ∞), (1.1)u(x, t) =, x ∈ ∂Ω, t ≥, (1.2)u(x, ) = u (x), ut(x, ) = u (x), x ∈ Ω, (1.3)where Ω ⊂ Rn (n ≥ ) is a bounded domain with a smooth boundary ∂Ω, ω ≥, μ > −ωλ, (1.4)λ being the rst eigenvalue of the operator −∆ under homogeneous Dirichlet boundary conditions
In the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, in nite time blow up of the solution with sub-critical initial energy
We study initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic source term utt − ∆u − ω∆ut + μut = u ln |u|, (x, t) ∈ Ω × (, ∞), (1.1)
Summary
We study initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic source term utt − ∆u − ω∆ut + μut = u ln |u|, (x, t) ∈ Ω × ( , ∞),. The undamped hyperbolic equation utt − ∆u = f (u),. Was introduced by D’Alembert [1] to model the propagation of waves along vibrating elastic string. By introducing the potential well, the global existence and nite time blow up of solution to (1.5) with E( ) < d were proved by Payne and Sattinger in [2], [3] respectively. The nonlinear wave equation with linear weak damping term was considered by Levine and Serrin [4] in abstract form. Wei Lian, College of Automation, Harbin Engineering University, People’s Republic of China *Corresponding Author: Runzhang Xu, College of Automation, Harbin Engineering University, People’s Republic of China and College of Mathematical Sciences, Harbin Engineering University, People’s Republic of China
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
