Extinction of solutions in parabolic equations with different diffusion operators. Issue 16 (10th December 2021) - Liu, Bingchen; Wang, Yuxi; Wang, Lu
ABSTRACT: In this paper, we study the evolution p, q -Laplacian equations u t = d i v ( | ∇ u | p − 2 ∇ u ) + u α ∫ Ω v m d x and v t = d i v ( | ∇ v | q − 2 ∇ v ) + v β ∫ Ω u n d x with 1< p, q <2, subject to homogeneous Dirichlet boundary conditions. If m n > ( p − 1 − α ) ( q − 1 − β ), there exist suitable initial data such that vanishing solutions exist. If m n < ( p − 1 − α ) ( q − 1 − β ), we find the explicit scopes of initial data such that the solutions could not vanish, which complete the corresponding classifications of solutions in Math. Methods Appl. Sci. 39 (2016) 1325–1335 and Appl. Math. Comp. 259 (2015) 587–595, respectively. For the critical case m n = ( p − 1 − α ) ( q − 1 − β ), the solutions vanish in finite time with small initial data.