On the existence theory for the nonlinear thermoelastic plate equation

Abstract

In this paper, we analyze the nonlinear thermoelastic plates, with Fourier heat conduction, and consider a polynomial-type nonlinearity. We first develop a theoretical analysis of the corresponding linear system to derive time decay estimates in $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ and $H^s({\mathbb{R}}^n)$. Then, using that set of decay estimates and controlling the nonlinearity, we prove the existence and uniqueness of local solutions with initial data $(u(0),u_t(0),\theta (0))=(u_0,\mathrm{\Delta }{u}_1,\mathrm{\Delta }{\theta }_1)$, with $u_0\in H^s$, and $u_1,{\theta }_1\in H^{s+1}$, for $s>\frac{n}{2}+1$.

Keywords:

• Thermoelastic plates
• Fourier heat conduction
• time decay rates
• local solutions

AMS subject classifications:

• 35A01
• 35G20
• 35L25
• 74B20
• 74K20

Acknowledgements

The first author has been supported by Universidad de Córdoba, Project FCB-05-22. The third author has been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander and Project CIPIF-MATH-AmSud de 2022. The authors would like to thank the anonymous referees for useful remarks and suggestions.

Disclosure statement

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