Global non-existence for some nonlinear wave equations with damping and source terms in an inhomogeneous medium

Yanjin Wang,Yufeng Wang

doi:10.1186/s13661-017-0762-4

Abstract

In this paper we study the initial value problem for the nonlinear wave equation with damping and source terms \t\t\tutt−ρ(x)−1Δu+ut+m2u=f(u)\\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}$$ u_{tt}-\\rho(x)^{-1}\\varDelta u+u_{t}+m^{2}u=f(u) $$\\end{document} with some rho(x) and f(u) on the whole space mathbb{R}^{n} (ngeq 3).For the low initial energy case, which is the non-positive initial energy, based on a concavity argument we prove the blow-up result. As for the high initial energy case, we give sufficient conditions of the initial data such that the corresponding solution blows up in finite time. In other words, our results imply a complete blow-up theorem in the sense of the initial energy, -infty< E(0)<+infty.

Highlights

In this paper our aim is to study some nonlinear wave equation with damping and source terms in the following form:⎧ ⎪⎨ utt – ρ(x)– Δu + ut + m u = f (u), (t, x) ∈ [, T) × Rn, ⎪⎩u(, x) = u (x), ut(, x) = u (x), x ∈ Rn, x ∈ Rn, ( )where Δ is Laplacian operator on Rn (n ≥ ), u (x) and u (x) are real valued functions, m is a real constant, ρ(x) satisfies the following condition:(H) ρ(x) > for every x ∈ Rn, ρ ∈ C,γ (Rn) with γ ∈ (, ), and ρ ∈ Ln/ (Rn) ∩ L∞(Rn)
Wang and Wang Boundary Value Problems (2017) 2017:39 words, ρ(x) = constant implies that the medium where the sound travels is inhomogeneous
We briefly introduce some results for the wave equation ( ) with ρ(x) = constant = (without loss of generality let ρ(x) = ), obviously it does not satisfy the assumption (H)

Summary

Introduction

We briefly introduce some results for the wave equation ( ) with ρ(x) = constant = (without loss of generality let ρ(x) = ), obviously it does not satisfy the assumption (H). The general nonlinear power f (u) with ( ) it was firstly considered for some abstract wave equations in [ ], where Levine proved the blow-up result when the initial energy was negative.

Objectives

Results

Conclusion