Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

الملخص EN

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions: (b(u))t=∇·(h(t)k(x)a(u)∇u)+f(x,u,|∇u|2,t), in D×(0,T), (∂u/∂n)+γu=0, on ∂D×(0,T), u(x,0)=u0(x)>0, in D¯, where D⊂RN (N≥2) is a bounded domain with smooth boundary ∂D.

Under some appropriate assumption on the functions f, h, k, b, and a and initial value u0, we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.