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

In this paper, we study the nonlocal nonlinear evolution equation ^CD_0|t ^αu(t,x)−(J∗|u|−|u|)(t,x)+^CD_0|t ^βu(t,x)=|u(t,x)|^p,t>0,x∈R^d,where 1<α<2, 0<β<1, p>1, J:R^d→R₊, ∗ is the convolution product in R^d, and ^CD_0|t ^q, 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 1CD_0|t ^αu(t,x)−(J∗|u|−|u|)(t,x)+^CD_0|t ^βu(t,x)=|v(t,x)|^p,t>0,x∈R^d,^CD_0|t ^αv(t,x)−(J∗|v|−|v|)(t,x)+^CD_0|t ^βv(t,x)=|u(t,x)|^q,t>0,x∈R^d,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