On the existence theory for the nonlinear thermoelastic plate equation

• Published in: Applicable analysis 2024-02, Vol.103 (3), p.636-656
• Main Authors: Banquet, Carlos, Doria, Mario, Villamizar-Roa, Élder J.
• Format: Article
• Language: English
• Subjects: Conduction heating; Conductive heat transfer; Fourier heat conduction; local solutions; Nonlinearity; Polynomials; Thermoelastic plates; time decay rates
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036811.2023.2202177

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^{\infty }(\mathbb {R}^n) $ L ∞ ( R n ) and $ H^s(\mathbb {R}^n) $ H s ( 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,\Delta u_1,\Delta \theta _1) $ ( u ( 0 ) , u t ( 0 ) , θ ( 0 ) ) = ( u 0 , Δ u 1 , Δ θ 1 ) , with $ u_0\in H^s $ u 0 ∈ H s , and $ u_1,\theta _1\in H^{s+1} $ u 1 , θ 1 ∈ H s + 1 , for $ s \gt \frac {n}{2}+1 $ s > n 2 + 1 .