Phase-lag heat conduction: decay rates for limit problems and well-posedness

Abstract

In two recent papers the authors have studied conditions on the relaxation parameters in order to guarantee the stability or instability of solutions for the Taylor approximations to dual-phase-lag and three-phase-lag heat conduction equations. However, for several limit cases relating to the parameters the kind of stability was unclear. Here we analyze these limit cases and clarify whether we can expect exponential or slow decay for the solutions. Moreover, rather general well-posedness results for three-phase-lag models are presented. Finally, the exponential stability expected by spectral analysis is rigorously proved exemplarily.

Spatial behavior in high‐order partial differential equations

“…In this section, we investigate the spatial behavior of the solutions of the problem determined by the systems (10) and (11) with the initial conditions (7) and (12) and the boundary conditions (8), (9), (13), and (14).…”

“…Theorem. Let us consider (u i , ) be a solution of the problem determined by the systems (10) and (11) with the initial conditions (7) and (12) and the boundary conditions (8), (9), (13), and (14). Then, for large enough, either the solution satisfies the condition (69) or it satisfies the decay estimates (72) and (76).…”

“…For this reason a big interest has been developed to understand different formal Taylor approximations to these equations. [12][13][14][15][16][17][18][19] In fact, the literature concerning these topics is increasing quickly because many researchers are interested in these proposals. These new theories allow to obtain stability of solutions and the well posedness of the problems, provided that certain conditions on the parameters hold.…”

Time decay in dual-phase-lag thermoelasticity: Critical case

“…To clarify the stability of this equation is the aim of many research works [1,19,21,15] (among others). It is known that the solutions of this equation decay exponentially when τ q < 2τ θ and polynomially when τ q = 2τ θ .…”

“…Theorem 3.1. The semigroup e tA is polynomially stable of order γ = 1 2 when 2τ θ = τ q , i.e., for all z 0 ∈ D(A), there is a constant C > 0 such that the solution z of (19) satisfies…”