Existence and blow up time estimate for a nonlinear Cauchy problem with variable exponents: theory and numerics

Abstract

A nonlinear Cauchy problem of a wave equation with variable exponents is considered. The damping term is subjected to a coefficient of both variables space and time. We prove the local existence of solutions using the Galerkin approach while the blow-up result is obtained via the perturbed energy method. We finally end our paper with some numerical illustrations. Problems that are frequently used in the literature are special cases.

Keywords:

• Blow-up
• finite time
• local existence
• variable exponents
• numerical computations

AMS Subject Classifications:

• 35A01
• 35B45
• 35L15
• 35L70
• 65M60

Disclosure statement

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

Additional information

Funding

The authors would like to express their sincere thanks to King Fahd University of Petroleum and Minerals. The authors also thank the referees for their very careful reading and valuable comments. This work was funded by KFUPM under Project No. # SB201026.