Blow-up analysis of solutions for weakly coupled degenerate parabolic systems with nonlinear boundary conditions. (October 2021) - Ding, Juntang
Abstract: This paper is devoted to the study of the blow-up solutions of the following weakly coupled degenerate parabolic systems with nonlinear boundary conditions: u t = div | ∇ u | p ∇ u + f ( x, u, v, t ), v t = div | ∇ v | q ∇ v + g ( x, u, v, t ), ( x, t ) ∈ D × ( 0, T ∗ ), ∂ u ∂ n = b ( u ), ∂ v ∂ n = d ( v ), ( x, t ) ∈ ∂ D × ( 0, T ∗ ), u ( x, 0 ) = u 0 ( x ), v ( x, 0 ) = v 0 ( x ), x ∈ D ¯ . Here p > 0, q > 0, D is a bounded spatial region in R n ( n ≥ 2 ), and the boundary ∂ D is smooth. We mainly combine the maximum principles of the weakly coupled parabolic systems with the first-order differential inequality technique to discuss the blow-up phenomenon of the above problem. Sufficient conditions for the blow-up of the nonnegative solution ( u, v ) of this problem are given. In addition, for the nonnegative blow-up solution ( u, v ), we also obtain an upper bound on the blow-up time and an upper estimate of the blow-up rate.