Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

Abstract

In this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.

Existence and asymptotic behaviour of solutions for the wave equation with a time varying delay and logarithmic nonlinearity

“…Especially, the energy decay results we obtained are optimal and can be displayed graphically by selecting suitable functional in dissipation term. By considering the interaction of time varying delay and the logarithmic source term, these results extend the earlier ones in the literature [5] and [14].…”

“…Furthermore, two types of energy decay results under different conditions are optimal. Our works improves the result in [5], [14] by considering the logarithmic nonlinearity or time varying delay. In our situation, we overcome the difficulty brought by the interaction of time varying delay and logarithmic nonlinearity.…”

“…By means of Faedo-Galerkin's scheme and Logarithmic Sobolev inequality, Park [14] showed the local and global solutions. Furthermore, through the perturbed energy method combined with potential-well method, the general decay rates and nonexistence of solutions was investigated.…”

“…Recently, Yüksekkaya [15] improved the earlier result of [14] by considering the term ∆ 2 u in (1.5) as follows:…”

On Behavior of Solutions for Nonlinear Klein–Gordon Wave Type Models with a Logarithmic Nonlinearity and Multiple Time-Varying Delays

“…A small delay can affect considerably dynamical behaviors of the system (e.g., destabilize the system which is asymptotically stable in the absence of time delays unless additional conditions, control functions, or stabilization mechanism functions have been used). Delay terms can lead to change in the stability of dynamics and give rise to highly complex behavior including instability, oscillations, and chaos (see, e.g., [1,[31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] and the references therein). Therefore, these behaviors and aspects, by taking into account different sources of delays, motivate the study of multiple time-varying delays effects on properties of dynamical systems.…”