In this note we consider the following linear partial different equation which is usually seen as an approximation to the dual-phase-lag heat equation proposed by Tzou. $$ \dot {T}+\tau_q \ddot {T}+\frac{\tau_{q}^{2}}{2}\dddot{T}=\kappa \triangle T+\kappa\tau_{T}\triangle \dot {T}+ \kappa \frac{\tau_{T}^{2}}{2}\triangle \ddot {T} $$ on a bounded domain $\Omega$ in $R^n$ with smooth boundary. We obtain analyticity for the associated $C_0-$semigroup. Moreover, we also obtain exponential stability of the solutions by spectrum analysis and Hurwitz criterion under one of the following conditions: (1) $\displaystyle{\frac{\tau_{T}}{\tau_{q}}>2-\sqrt{3}};$ (2) $2-\sqrt{3} \ge \displaystyle{\frac{\tau_{T}}{\tau_{q}} >\frac{\sqrt{(1+\kappa\tau_{T}\lambda_1)^{2}+(\kappa\tau_{T}\lambda_1)^{2}+(\kappa\tau_{T}\lambda_1)^{3}}-(1+\kappa\tau_{T}\lambda_1)}{\kappa\tau_{T}\lambda_{1}(1+\kappa\tau_{T}\lambda_1)}},$ where $\lambda_1$ is the smallest eigenvalue of the negative Laplacian on $\Omega$ with Dirichlet boundary condition.

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