Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity-II

ABSTRACT

This paper deals with the following mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity $u_t-\mathrm{\Delta }{u}_t-\mathrm{d}\mathrm{i}\mathrm{v}\left(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\right)=|u{|}^{q-2}u\log |u|$ 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) $1<p\le q\left\{\begin{array}{ll}\le 2,& \mathrm{i}\mathrm{f}n\le p,\\ \le 2,& \mathrm{i}\mathrm{f}\frac{2n}{n+2}<p<n,\\ <\frac{np}{n-p},& \mathrm{i}\mathrm{f}p\le \frac{2n}{n+2},\end{array}\right.$(1) the global existence and blow-up results were got when $J\left(u_0\right)\le d$, where d denotes the mountain-pass level. But for the case (2) $1<p\le q\mathrm{a}\mathrm{n}\mathrm{d}2<q<\left\{\begin{array}{ll}\mathrm{\infty },& \mathrm{i}\mathrm{f}n\le p,\\ \frac{np}{n-p},& \mathrm{i}\mathrm{f}\frac{2n}{n+2}<p<n,\end{array}\right.$(2) the blow-up results were got when $J\left(u_0\right)\le M$, where $M\le d$ 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 $J\left(u_0\right)\le d$ 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 $J\left(u_0\right)>d$.

Keywords:

• Pseudo-parabolic equation
• p-Laplacian
• upper and lower bounds of blow-up time
• high initial energy

2010 Mathematics Subject Classifications:

• 35K70
• 35B05
• 35B40

Disclosure statement

No potential conflict of interest was reported by the authors.