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=Δu+f(x,u,|∇u|2,t)inD×(0,T),∂u∂n=r(u)on∂D×(0,T),u(x,0)=u0(x)>0in D¯, where D⊂RND⊂RN is a bounded domain with smooth boundary ∂D∂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,rf,g,r, and initial value u0u0.