Global existence and blow-up problems for reaction diffusion model with multiple nonlinearities

In this paper, we study the following reaction diffusion model, ⎧⎪⎪⎨ ⎪⎪⎩ ut =∇ · a(u)b(x)c(t)∇u +g(x, t)f (u), in D ×(0,T ), ∂u ∂n = h(x, t)r(u), on ∂D ×(0,T ), u(x, 0) = u0(x) > 0, in ¯ D, where D is a bounded domain in RN with smooth boundary ∂D, N 2. This paper deals with interactions among three kinds of nonlinear mechanisms: nonlinear reaction, nonlinear convection and nonlinear boundary flux. The existence theorems of blowup positive solutions, upper bound of “blow-up time,” upper estimates of “blow-up rate,” existence theorems of global positive solutions, and upper estimates of global positive solutions are given under suitable assumptions on a, b, c,f, g,h, r and initial data u0(x). © 2008 Elsevier Inc. All rights reserved.