Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations

Abstract

In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate Equation with homogeneous Dirichlet boundary conditions), x ∈ Ω are continuous and locally Lipschitz functions. First, we prove that the linear system (f 1 = f 2 = 0) generates an analyitic strongly continuous semigroups which decays exponentially to zero. Second, under some additional condition we prove that the non-linear system has a bounded solution which is exponentially stable, and for a large class of functions f 1 , f 2 this bounded solution is almost periodic. Finally, we use the analyticity of the semigroup generated by the linear system to prove the smoothness of the bounded solution.

Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations

“…The fact that each operator A(t) be sectorial was shown in [21], however here we give a complete proof, as we have to determine the precise constants in order to comply with assumption (H1). Finally, by applying the abstract result developed in the previous section, we prove that the thermoelastic system (1.1) has a unique almost periodic solution u θ in H 1+α × H α .…”

“…For technical need, we assume furthermore that To show (2.1) appearing in (H1), we follow along the same lines as in [21]. For that, let 0 < λ 1 < λ 2 < · · · < λ n → ∞ be Hence, for z := w v θ ∈ D( A(t)), we have …”

“…It is worth mentioning that this question was recently studied by H. Leiva et al [21] in the case when not only the coefficients a, b were constant but also there was no gradient terms in the semilinear terms f 1 and f 2 .…”

Nonexistence results for the Cauchy problem for some fractional nonlinear systems of thermo‐elasticity type

“…and ξ (t) = φ γ t 2 T 2 with T > 0 and γ >> 1. By changing the variable of integration t = T τ , we get Now, by letting T −→ +∞, we obtain a contradiction as the second hand side goes to zero while I (0) + J (0) + γ λ 1 J (0) > 0 (see (13)), the left hand side is strictly positive.…”

Pseudo-Almost Automorphic Solutions to Some Second-Order Differential Equations

“…The existence of almost periodic solutions to second-order differential equations constitutes one of the most important topics in qualitative theory of differential equations due essentially to their applications such thermoelastic plate equations [12,37] or telegraph equation [43] or Sine-Gordon equations [36]. Some contributions on the maximal regularity, bounded, almost periodic, asymptotically almost periodic solutions to abstract second-order differential and partial differential equations have recently been made, among them are [9], [10], [18], [20], [29], [30], [39], [40], [52], [53], [54], [55], and [56].…”