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

Abstract

We consider the system of nonlinear wave equations with nonlinear time fractional damping utt+−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,where u,v=ut,x,vt,x, m and N are 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 order κ. 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.