Global existence and blow-up for a nonlinear porous medium equation

Abstract In this paper, we investigate the positive solution of nonlinear nonlocal porous medium equation u t −Δu m = au p ∝ Ω u q dx with homogeneous Dirichlet boundary condition and positive initial value u0(x), where m > 1, p, q ≥ 0. Under appropriate hypotheses, we establish the local existence and uniqueness of a positive classical solution, and obtain that the solution either exists globally or blows up in finite time by utilizing sub and super solution techniques. Furthermore, we yield the blow-up rate, i.e., there exist two positive constants C1, C2 such that where p+q > m > 1, T ∗ is the blow-up time of u(x,t).