Blow-up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions

In this paper, we consider the following reaction diffusion systems with gradient nonlinearity under nonlinear boundary condition {ut=△u+upvq−∣∇u∣α,(x,t)∈Ω×(0,t∗);vt=△v+vrus−∣∇v∣α,(x,t)∈Ω×(0,t∗);∂u∂ν=g(u),∂v∂ν=h(v),(x,t)∈∂Ω×(0,t∗);u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω where Ω⊂RN(N≥1) is a bounded region with smooth boundary ∂Ω, p,q,r,s≥0, α>1, t∗ is a possible blow-up time when blow-up occurs. By constructing an appropriate auxiliary functions, and by means of Payne-Weinberger or Scott's method, a lower bound on blow-up time when blow-up occurs is derived.