Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions

In this paper, we consider the following reaction diffusion equations under Dirichlet boundary condition{(g(u))t=∇⋅(ρ(|∇u|2)∇u)+k(t)f(u)in Ω×(0,t⁎),u(x,t)=0on ∂Ω×(0,t⁎),u(x,0)=u0(x)≥0in Ω‾, where Ω is a bounded domain of RNRN(N≥2)(N≥2) with smooth boundary ∂Ω. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, the upper and lower bounds on blow-up time when blow-up occurs are presented. Moreover, conditions on the data to ensure that the solution exists globally or blows up at some finite time are also derived.