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

Sun-Hye Park

doi:10.1186/s13662-020-03037-6

Abstract

In this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form utt(x,t)−Δu(x,t)+αut(x,t)+βut(x,t−τ)=u(x,t)ln|u(x,t)|γ.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}(x,t) - \\Delta u (x,t) + \\alpha u_{t} (x,t) + \\beta u_{t} (x, t- \\tau ) = u(x,t) \\ln \\bigl\\vert u(x,t) \\bigr\\vert ^{\\gamma } . $$\\end{document} 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.

Highlights

We consider the following wave equation with frictional damping, time delay in the velocity, and logarithmic source: utt – u + αut(t) + βut(x, t – τ ) = u ln |u|γ for (x, t) ∈ × (0, ∞), (1.1)u(x, t) = 0 for (x, t) ∈ ∂ × (0, ∞), (1.2)u(x, 0) = u0(x), ut(x, 0) = u1(x) for x ∈, (1.3)ut(x, t) = j0(x, t) for (x, t) ∈ × (–τ, 0), (1.4)where ⊂ RN, N ≥ 1, is a bounded domain with smooth boundary ∂ . τ > 0 is time delay, α, β, and γ are real numbers that will be specified later
For the strongly damped wave equation utt – u – a ut + but = u ln |u|γ, Ma and Fang [22] showed the global existence and infinite time blow-up of solutions when γ = 2, a = 1, and b = 0. They used a family of potential wells that is related to the logarithmic nonlinearity, which was introduced by Chen et al [7]
4 Global existence and energy decay estimate we prove the global existence and energy decay rates of solutions to problem (3.2)–(3.6)

Summary

Introduction

For the strongly damped wave equation utt – u – a ut + but = u ln |u|γ , Ma and Fang [22] showed the global existence and infinite time blow-up of solutions when γ = 2, a = 1, and b = 0. They used a family of potential wells that is related to the logarithmic nonlinearity, which was introduced by Chen et al [7].

This gives

Proof We set

Then we have from