Abstract
In this paper, we study the nonlocal nonlinear evolution equation
(C)D(0 vertical bar t)(alpha)u(t, x) - (J * vertical bar u vertical bar - vertical bar u vertical bar)(t, x) + D-C(0 vertical bar t)beta u(t, x) = vertical bar u(t, x)vertical bar(p), t > 0, x is an element of R-d,
where 1 < alpha < 2, 0 < beta < 1, p > 1, J : R-d -> R+, * is the convolution product in R-d, and D-C(0 vertical bar t)q, q is an element of {alpha, beta}, 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 + 2 beta/d beta+2(1-beta). Next, we deal with the system
{(C)D(0 vertical bar t)(alpha)u(t, x) - (J * vertical bar u vertical bar-vertical bar u vertical bar)(t, x) + (C)D(0 vertical bar t)(beta)u(t, x) = vertical bar v(t, x)vertical bar(p), t > 0, x is an element of R-d,
(C)D(0 vertical bar t)(alpha)v(t, x) - (J* vertical bar v vertical bar-vertical bar v vertical bar)(t, x) + (C)D(0 vertical bar t)(beta)v(t,x) = vertical bar u(t, x)vertical bar(q), t > 0, x is an element of R-d,
where 1 < alpha < 2, 0 < beta < 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+2 beta/d beta+2(1-beta)max{p + 1,q + 1}. Our approach is based on the test function method. (C) 2018 Elsevier Ltd. All rights reserved.