Global existence of solutions of a semilinear heat equation with nonlinear memory condition

Abstract

We consider a semilinear parabolic equation with flux at the boundary governed by a nonlinear memory. We give some conditions for this problem which guarantee global existence of solutions as well as blow-up in finite time of all nontrivial solutions. The results depend on the behavior of variable coefficients as $t\to \mathrm{\infty }.$

COMMUNICATED BY:

• Alexander Pankov

Keywords:

• Semilinear parabolic equation
• memory boundary condition
• finite time blow-up

2010 Mathematics Subject Classifications:

• 35K20
• 35K58
• 35K61

Acknowledgments

The first author is grateful to the University of Picardie Jules Verne, since a part of this work was done while he enjoyed the hospitality of this university.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by the “RUDN University Program 5-100” and the State Program of Fundamental Research of Belarus (grant 1.2.03.1). The second author was supported by DAI-UPJV F-Amiens.