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A Well-Posed Problem for the Three-Dual-Phase-Lag Heat Conduction

The three-dual-phase-lag theory based on the constitutive law q(P, t + τ q ) = −(k∇T(P, t + τ T ) + k*∇ν(P, t + τν)), was proposed by Roy Choudhuri as an extension of the theory by Tzou where the recent theories of Greeen and Naghdi could be recover. Although it proposes a law that could be compatible with our intuition, when we adjoin it with the energy equation −∇q(x, t) = cṪ(x, t), we obtain an ill-posed problem, that is, a problem which has a sequence of eigenvalues such that their real part are positive (and go to infinity). A consequence of this fact is that the problem is always unstable and we cannot obtain continuous dependence on the initial data. In this note we combine this constitutive equation with two-temperature heat conduction theory and we show in this new context that the problem is well-posed.