The Cauchy–Dirichlet problem for the Moore–Gibson–Thompson equation

Abstract

The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation is analyzed. With the focus on non-homogeneous boundary data, two approaches are offered: one is based on the theory of hyperbolic equations, while the other one uses the theory of operator semigroups. This is a mixed hyperbolic problem with a characteristic spatial boundary. Hence, the regularity results exhibit some deficiencies when compared with the non-characteristic case.Résumé. On analyse le problème de Cauchy-Dirichlet pour l'équation de Moore-Gibson-Thompson avec des données non-homogènes. Deux méthodes sont considérées: la théorie des équations hyperboliques et la théorie des semi-groupes d'opérateurs. Il s'agit d'un problème hyperbolique mixte avec une frontière spatiale caractéristique. Par conséquent, les résultats de régularité présentent certaines lacunes par rapport au cas non caractéristique.

From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

“…Thus the present paper may be viewed as a corollary of this observation and the results in [17] for 2nd order hyperbolic equations in the Dirichlet case; and in [22], [23], [37] in the Neumann case. By contrast, reference [3] employs two distinct approaches: one based on Sakamoto's theory of hyperbolic systems [33,34,29,31] for the case g ∈ H 2 (Σ) and one which exploits the connection of the third-order problem to a wave equation with memory, an idea proposed in [2]. In turn, such second approach of [3] critically appeals to the hyperbolic regularity theory for linear wave equation with Dirichlet control as in [17, Theorem 3.8, p. 180-182; Theorem 2l2, p. 152], which is recalled in our Theorem 3.2.1.…”

“…More importantly, as to the boundary regularity, our Theorem 3.2.2 Part A (ii) [same as for the wave equation case [17,Theorem 2.2,p. 152]] establishes ∂y ∂ν Σ ∈ H 1 (Σ) under g ∈ H 2 (Σ), thus answering in the positive a question raised in [3,Remark 2(iii)].…”

“…In a first comparison with [3, Theorem 1(a)], we note the following differences regarding the interior and the boundary regularity. As to the interior regularity, our Theorem 3.2.2 Part A(i) is slightly more precise than [3,Theorem 1(a)] in that the additional assumption g t ∈ C(0, T ; H 1 2 (Γ)) in [3,Eq. (9)] is needed only for y t , y tt ∈ C(0, T ; H 1 (Ω) × L 2 (Ω)), but not for y ∈ C(0, T ; H 2 (Ω)).…”

“…1. For comparison purposes, the only reference in the literature of the third order problem with a boundary control g "smoother" than g ∈ L 2 (0, T ; L 2 (Γ)) is [3], which considers only the case where g is a Dirichlet control. [The complementary case g = 0 was studied on several function spaces in [30], see Appendix B, for both the Dirichlet or the Neumann case.]…”

“…In turn, such second approach of [3] critically appeals to the hyperbolic regularity theory for linear wave equation with Dirichlet control as in [17, Theorem 3.8, p. 180-182; Theorem 2l2, p. 152], which is recalled in our Theorem 3.2.1. As noted in [3,Introduction]:…”