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),inD×(0,T),∂u∂n=h(x,t)r(u),on∂D×(0,T),u(x,0)=u0(x)>0,inD¯, where D   is a bounded domain in RNRN with smooth boundary ∂D  , N⩾2N⩾2. This paper deals with interactions among three kinds of nonlinear mechanisms: nonlinear reaction, nonlinear convection and nonlinear boundary flux. The existence theorems of blow-up 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,ra,b,c,f,g,h,r and initial data u0(x)u0(x).