Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity-II. Issue 12 (10th September 2021) - Ding, Hang; Zhou, Jun
ABSTRACT: This paper deals with the following mixed pseudo-parabolic p -Laplacian type equation with logarithmic nonlinearity 1 u t − Δ u t − d i v ( | ∇ u | p − 2 ∇ u ) = | u | q − 2 u log ⁡ | u | in a bounded domain with zero Dirichlet boundary condition, which was studied in our previous paper [J Math Anal Appl. 2019;478(2):393-420]. In view the results of [J Math Anal Appl. 2019;478(2):393-420], for the case (1) 1 < p ≤ q ≤ 2, i f   n ≤ p, ≤ 2, i f   2 n n + 2 < p < n, < n p n − p, i f   p ≤ 2 n n + 2, the global existence and blow-up results were got when J ( u 0 ) ≤ d, where d denotes the mountain-pass level. But for the case (2) 1 < p ≤ q   a n d   2 < q < ∞, i f   n ≤ p, n p n − p, i f   2 n n + 2 < p < n, the blow-up results were got when J ( u 0 ) ≤ M, where M ≤ d is a constant. In this paper, we extend and complete the results of [J Math Anal Appl. 2019;478(2):393-420] on the following three aspects: First, the blow-up results are got when J ( u 0 ) ≤ d and (2) are satisfied. Second, the upper and lower bounds of blow-up time are estimated. Third, the global existence and blow-up results are got when J ( u 0 ) > d .