Blow-up and global solutions for nonlinear reaction–diffusion equations with Neumann boundary conditions

The type of problem under consideration is {((1+u)lnα(1+u))t=∇⋅(lnσ(1+u)∇u)+(1+u)lnβ(1+u),in D×(0,T),∂u∂n=0,on ∂D×(0,T),u(x,0)=u0(x)>0,in D̄, where D⊂RND⊂RN is a bounded domain with smooth boundary ∂D∂D, N≥2N≥2. It is proved that if β−1>σ≥α≥0β−1>σ≥α≥0, the positive solution u(x,t)u(x,t) blows up globally in D̄, whereas if 0≤β≤σ≤α−10≤β≤σ≤α−1, the positive solution u(x,t)u(x,t) is global solution. Furthermore, an upper bound of the “blow-up time”, an upper estimate of the “blow-up rate”, and an upper estimate of the global solutions are given.