Nonexistence of Global Weak Solutions of a System of Nonlinear Wave Equations with Nonlinear Fractional Damping

Mohamed Jleli,Bessem Samet,Mokhtar Kirane

doi:10.1155/2020/5764195

Abstract

We consider the system of nonlinear wave equations with nonlinear time fractional dampingutt+−Δmu+CD0,tαtσuq=vp,t>0,x∈ℝN,vtt+−Δmv+CD0,tβtδvr=vs,t>0,x∈ℝN,u0,x,ut0,x=u0x,u1x,x∈ℝN,u0,x,ut0,x=u0x,u1x,x∈ℝN,whereu,v=ut,x,vt,x,mandNare positive natural numbers,p,q,r,s>1,σ,δ≥0,0<α,β<1, and CD0,tκ,0<κ<1, is the Caputo fractional derivative of orderκ. Namely, sufficient criteria are derived so that the system admits no global weak solution. To the best of our knowledge, the considered system was not previously studied in the literature.

Highlights

In this paper, we investigate the system of nonlinear wave equations with nonlinear time fractional damping: 8 >>>>>> > 0, 0, x x >>>>>>: u1 u1ðxÞÞ, ðxÞÞ, x x ð1Þ where ðu, vÞ = ðuðt, xÞ, vðt, xÞÞ, m and N are positive natural numbers, p, q, r, s > 1, σ and δ are nonnegative numbers that will be specified later, 0 < α, β < 1, and CDκ0,t, 0 < κ < 1, is the Caputo fractional derivative of order κ
We are interested in obtaining sufficient conditions for which the considered system admits no global weak solution
Our approach is based on the nonlinear capacity method

Summary

Introduction

We investigate the system of nonlinear wave equations with nonlinear time fractional damping:. We are interested in obtaining sufficient conditions for which the considered system admits no global weak solution. If u is a solution to (2), there exist T∗ ≤ ∞ and sufficiently large. It was shown that the solution of (2) is unbounded and grows up exponentially in the Lp+1 -norm for sufficiently large initial data.

Preliminaries

Main Result

Proof of Theorem 4