Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

Abstract

We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established for a quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple dimensions.

Boundary stabilization of quasilinear Maxwell equations

“…When studying a quasilinear dynamics in bounded domains, it is commonly recognized that the global well-posedness goes hand in hand with the exponential stability of classical solutions (cf. [30,31,32,38], etc.). Hence, a big part of the present paper is devoted to showing the desired stability property.…”

“…Using Young's inequality to estimate z 3/2 (τ ) ≤ 1 2 z(τ ) + z 2 (τ ) , the estimate in (4.3) yields the desired inequality with an appropriate positive constant C. Theorem 2.2 can now easily be proved, cf. [30] or [31].…”

“…In contrast to semilinear problems, uniform stability is often the only way to global existence, especially for hyperbolic problems. In bounded domains, it is the exponential stability that furnishes the global existence of smooth solutions originating in a vicinity of a stable equilibrium [21,30,31,32,38,42]. In the full space R d , dispersive estimates are the path to the long-time existence [36,37,40,41,49,50,56].…”

“…Then our main result from Theorem 2.2 is a direct consequence of the standard barrier method (cf. [30,31,32,38,42], etc. ).…”

Nonlinear thermoelastic plate equations – Global existence and decay rates for the Cauchy problem

“…Extensions to more general nonlinearities, considering > 0, were given by Lasiecka, Pokojovy and Wan in [8], still in bounded domains. Semilinear problems for the Cauchy problem have been successfully treated by Fischer [6].…”