Lower and upper bounds for the blow-up time to a viscoelastic Petrovsky wave equation with variable sources and memory term

ABSTRACT

Under Dirichlet boundary conditions, we consider here a new type of viscoelastic Petrovsky wave equation involving variable sources and memory term $u_{tt}+{\mathrm{\Delta }}^{2}u-{\int }_0^{t}g(t-s){\mathrm{\Delta }}^{2}u(x,s)\mathrm{d}s+{\left|u_t\right|}^{m(.)-2}{u}_t={\left|u\right|}^{p(.)-2}u.$We discuss the blow-up in finite time with arbitrary positive initial energy and suitable large initial values if $p(.)$ and the relaxation function g satisfies some conditions. Employing a different method for higher bounded positive initial energy, not only finite time blow-up for solutions proved but also the lower and upper bounds for blowing up time are gotten.

KEYWORDS:

• Nonlinear Petrovsky equation
• memory
• lower and upper bounds for the blow-up
• critical exponents
• variable nonlinearity

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35L70
• 35B44
• 74D10

Acknowledgments

The author would like to thank the anonymous referee(s) and the handling editor(s) for their kind comments.

Disclosure statement

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