Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

Abstract In this paper we discuss the blow-up for classical solutions to the following class of parabolic equations with Robin boundary condition: { ( b ( u ) ) t = ∇ ⋅ ( g ( u ) ∇ u ) + f ( u ) in  Ω × ( 0 , T ) , ∂ u ∂ n + γ u = 0 on  ∂ Ω × ( 0 , T ) , u ( x , 0 ) = h ( x ) ≥ 0 in  Ω ¯ , where Ω is a bounded domain of R N ( N ≥ 2 ) with smooth boundary ∂ Ω . By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, we derive conditions on the data which guarantee the blow-up or the global existence of the solution. For the blow-up solution, a lower bound on blow-up time is also obtained. Moreover, some examples are presented to illustrate the applications.