Pseudoparabolic equations with variable exponents and coefficients: blow-up and large time behaviors

ABSTRACT

In this paper, the following nonlinear generalized pseudoparabolic equation with variable exponents and coefficients is considered: $\begin{array}{rl}& u_t-\mathrm{div}(\alpha (x,t){\left|\mathrm{\nabla }u\right|}^{\sigma (x)-2}\mathrm{\nabla }{u}_t+\beta (x,t){\left|\mathrm{\nabla }u\right|}^{p(x)-2}\mathrm{\nabla }u)\\ & =\gamma (x,t){\left|u\right|}^{m(x)-2}u.\end{array}$Under suitable assumptions on the specified initial data and the variable exponents q, p, m, and the coefficients $\alpha ,\beta ,\gamma$, the finite-time blow-up and large time behavior (exponential and power decay) of the weak solutions are established.

KEYWORDS:

• Pseudoparabolic
• blow-up
• large time behavior
• variable exponents
• variable coefficients

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35K70
• 35B44
• 35B40
• 35N10

Acknowledgments

The author is very grateful to the referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the grant of the Ministry of Education and Science of the Republic of Kazakhstan (MES RK), Kazakhstan [Grant no. AP08052425].