Global existence and blow-up solutions for quasilinear reaction–diffusion equations with a gradient term

In this work, we study the blow-up and global solutions for a quasilinear reaction–diffusion equation with a gradient term and nonlinear boundary condition: { ( g ( u ) ) t = Δ u + f ( x , u , | ∇ u | 2 , t ) in D × ( 0 , T ) , ∂ u ∂ n = r ( u ) on ∂ D × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) > 0 in  D ¯ , where D ⊂ R N is a bounded domain with smooth boundary ∂ D . Through constructing suitable auxiliary functions and using maximum principles, the sufficient conditions for the existence of a blow-up solution, an upper bound for the “blow-up time”, an upper estimate of the “blow-up rate”, the sufficient conditions for the existence of the global solution, and an upper estimate of the global solution are specified under some appropriate assumptions on the nonlinear system functions f , g , r , and initial value u 0 .