On a nonlinear heat equation with viscoelastic term associated with Robin conditions

Abstract

This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions.

Keywords:

• Faedo–Galerkin method
• Nonlinear heat equation; Robin conditions; Blow up; Asymptotic behavior of solutions

AMS Subject Classifications:

• 34B60
• 35K55
• 35Q72
• 80A30

Acknowledgements

The authors wish to express their sincere thanks to the Editor and the referees for the suggestions, remarks and valuable comments.

Notes

No potential conflict of interest was reported by the authors.