Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source

We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source u(t)+(-Delta)beta/2 u=(1+vertical bar x vertical bar)gamma integral 0t(t-s)alpha-1 vertical bar u vertical bar p parallel to nu 1q(x)u parallel to(r)(q)ds for (x,t) is an element of R(N)x(0,infinity) with initial data u(x,0)=u(0)(x)is an element of L-loc(1)(R-N), where p,q,r > 1, where p, q, r > 1, q(p+r)q+r, 0 gamma <= 2 , 0 alpha 1, 0 beta <= 2, (-Delta)beta 2 stands for the fractional Laplacian operator of order beta, the weight function nu(x) is positive and singular at the origin, and parallel to.parallel to(q) is the norm of L-q space.