On the phase-lag heat equation with spatial dependent lags

In this paper we investigate several qualitative properties of the solutions of the dual-phase-lag heat equation and the three-phase-lag heat equation. In the first case we assume that the parameter τ_T depends on the spatial position. We prove that when 2τ_T−τ_q is strictly positive the solutions are exponentially stable. When this property is satisfied in a proper sub-domain, but 2τ_T−τ_q≥0 for all the points in the case of the one-dimensional problem we also prove the exponential stability of solutions. A critical case corresponds to the situation when 2τ_T−τ_q=0 in the whole domain. It is known that the solutions are not exponentially stable. We here obtain the polynomial stability for this case. Last section of the paper is devoted to the three-phase-lag case when τ_T and τ_ν ^⁎ depend on the spatial variable. We here consider the case when τ_ν ^⁎≥κ^⁎τ_q and τ_T is a positive constant, and obtain the analyticity of the semigroup of solutions. Exponential stability and impossibility of localization are consequences of the analyticity of the semigroup.