Abstract
We investigate the finite time blow-up and global existence of sign-changing solutions to the Cauchy problem for the inhomogeneous semilinear parabolic system with space-time forcing terms
{u(t) - Delta u = vertical bar v vertical bar(p) + t(sigma)w(1)(x), x is an element of R-N, t > 0, v(t) - Delta v = vertical bar u vertical bar(q) + t(gamma)w(2)(x), x is an element of R-N, t > 0, (u(0, x), v(0, x) = (u(0)(x), v(0)(x)), x is an element of R-N,
where N >= 1, p, q > 1, sigma, gamma > -1, sigma, gamma not equal 0, w(1), w(2) not equivalent to 0, and u(0), v(0) is an element of C-0(R-N). For the finite time blow-up, two cases are discussed under the conditions w(i) is an element of L-1(R-N) and integral(RN) w(i)(x) dx > 0, i = 1, 2. Namely, if sigma > 0 or gamma > 0, we show that the (mild) solution (u, v) to the considered system blows up in finite time, while if sigma, gamma is an element of (-1, 0), then a finite time blow-up occurs when N/2 < max {(sigma+1)(pq-1)+p+1/pq-1, (gamma+1)(pq-1)+q+1/pq-1}. Moreover, if N/2 >= max {(sigma+1)(pq-1)+p+1/pq-1, (gamma+1)(pq-1)+q+1/pq-1}, p > sigma/gamma and q > gamma/sigma, we show that the solution is global for suitable initial values and w(i), i = 1, 2. (C) 2021 Elsevier Ltd. All rights reserved.