Remarks on some dissipative sine-Gordon equations

ABSTRACT

In this short note, we discuss some results concerning the long-time behaviour of solutions to dissipative sine-Gordon equation in a bounded domain of ${\mathbb{R}}^n$. We first prove that without forcing term and under some assumptions on the domain then the global attractor reduces to $\left\{0\right\}$. We then rewrite the sine-Gordon equation as a Schrödinger equation with smoothing; this allows us to provide another proof for the so-called asymptotical smoothing effect for this dissipative wave equation.

COMMUNICATED BY:

• V. Rdulescu

KEYWORDS:

• Sine-Gordon equations
• attractors
• solutions of elliptic equations

AMS SUBJECT CLASSIFICATIONS:

• 35B41
• 37L50
• 35L71

Acknowledgments

The author would like to thank Alain Miranville and Morgan Pierre for fruitful discussions.

Disclosure statement

No potential conflict of interest was reported by the author.