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 = ∆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 .