Nonexistence of global solutions for a class of nonlocal in time and space nonlinear evolution equations

Abstract

In this paper, we study the nonlocal nonlinear evolution equation CD0|tαu(t,x)−(J∗|u|−|u|)(t,x)+CD0|tβu(t,x)=|u(t,x)|p,t>0,x∈Rd,where 1<α<2, 0<β<1, p>1, J:Rd→R+, ∗ is the convolution product in Rd, and CD0|tq, q∈{α,β}, is the Caputo left-sided fractional derivative of order q with respect to the time t. We prove that the problem admits no global weak solution other than the trivial one with suitable initial data when 1<p<1+2βdβ+2(1−β). Next, we deal with the system CD0|tαu(t,x)−(J∗|u|−|u|)(t,x)+CD0|tβu(t,x)=|v(t,x)|p,t>0,x∈Rd,CD0|tαv(t,x)−(J∗|v|−|v|)(t,x)+CD0|tβv(t,x)=|u(t,x)|q,t>0,x∈Rd,where 1<α<2, 0<β<1, p>1, and q>1. We prove that the system admitsnon global weak solution other than the trivial one with suitable initial data when 1<pq<1+2βdβ+2(1−β)max{p+1,q+1}. Our approach is based on the test function method.