On a nonlinear boundary problem for thermoelastic coupled beam equations with memory term

Abstract

In this paper, we study a one-dimension nonlinear problem for thermoelastic coupled beam equations with memory term. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, we first prove the local existence and the uniqueness of a weak solution. Next, by establishing assumptions and constructing energy functionals suitably, we consider the global existence and general decay behavior of the solution. Finally, the blow-up property in the special case of this problem is also given. KEYWORDSthermoelastic coupled beam equations, Faedo-Galerkin method, linearization method, local existence and global existence, general decay, blow-up MSC CLASSIFICATION35A01, 35M33, 35B40, 35B44 J. A. Burns et al 9 proved the exponential decay energy for a linear homogeneous thermoelastic bar in length 𝜋 with unit reference density subject to various boundary conditions. C. M. Dafermos 10 proved on the existence, differentiability, and asymptotic stability of solutions to a system of linear thermoelasticity. For a model of linear non-Fourier thermoelastic bar, C. Giorgi and M. G. Naso 11 proved the exponential stability of c 0 -semigroup associated with the corresponding system. J. E. Muñoz Rivera 12 established the decay rate of energy in one-dimensional linear thermoelasticity obeying Fourier's law without memory effect. Afterward, J.E. Muñoz Rivera and Y. Qin 13 studied the global existence, uniqueness, and asymptotic behavior of solutions to the equations of one-dimensional nonlinear thermoelasticity with thermal memory subject to Dirichlet-Dirichlet boundary conditions at the endpoints. Slemrod 14 proved the global existence and asymptotic stability of small solutions with Neumann-Dirichlet or Dirichlet-Neumann boundary conditions.In the following, let us mention to the results in some previous works for the long-time dynamics of thermoelastic coupled beam system with thermal effects. Giorgi et al 15 studied the dissipative system { u tt + Δ 2 u + Δ𝜃 − (𝛽 + ‖u x (t)‖ 2 )Δu = 𝑓 , 𝜃 t − Δ𝜃 − Δu t = g,

Global Non-existence of a Coupled Parabolic-Hyperbolic System of Thermoelastic Type with History

“…Similar problems have been studied in [6][7][8], and in [9,10] where a rotational term is considered.…”

Global Non-Existence of a Coupled Parabolic–Hyperbolic System of Thermoelastic Type with History

“…Furthermore, for every positive value of the initial energy, there exist initial data, such that the corresponding solution blows up. Problems similar to (ThE) * 2 have studied long-time dynamics and decay estimates of the solution to zero, in [6,27,28]. However, to our knowledge, our analysis is the first to address the blow-up.…”

“…See [6][7][8] for the physics of the model. These sources point out that this problem considers a heat flux theory due to the Coleman-Gurtin with parameter ω ∈ (0, 1).…”