Blow-up phenomenon for a semilinear pseudo-parabolic equation involving variable source

Abstract

We consider a semilinear pseudo-parabolic equation with nonlinearities of variable exponent subject to Dirichlet boundary conditions $u_t-\mathrm{\Delta }{u}_t-\mathrm{\Delta }u=u^{p\left(x\right)}.$We prove that the nonnegative classical solutions blow-up in finite time with arbitrary positive initial energy and suitable large initial values. Also, employing a differential inequality technique, we obtain an upper bound for blow-up time if $p(.)$ and the initial value satisfies some conditions.

Keywords:

• Blow-up
• lower bound
• semilinear pseudo-parabolic equations
• Sobolev spaces with variable exponents
• variable nonlinearity

2010 Mathematics Subject Classifications:

• 35B44
• 35K20

Acknowledgments

The author would like to thank the anonymous referee(s) and the handling editor(s) for fruitful comments through the significant improvement of this paper.

Disclosure statement

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