Blow-up and global existence for nonlinear parabolic equations with Neumann boundary conditions

In this paper, we study the global and blow-up solutions of the following problem: {(h(u))t=∇⋅(a(u,t)b(x)∇u)+g(t)f(u)in D×(0,T),∂u∂n=0on ∂D×(0,T),u(x,0)=u0(x)>0in D¯, where D⊂RND⊂RN is a bounded domain with smooth boundary ∂D∂D. By constructing auxiliary functions and using maximum principles and comparison principles, the sufficient conditions for the existence of global solution, an upper estimate of the global solution, the sufficient conditions for the existence of the blow-up solution, an upper bound for the “blow-up time”, and an upper estimate of the “blow-up rate” are specified under some appropriate assumptions on the functions a,b,f,ga,b,f,g, and hh.