Boundary value problems for Sobolev type equations with irreversible operator coefficient of the highest derivatives

A. I. Kozhanov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the IV International Scientific Conference "Actual Problems of Applied Mathematics". Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part III, Tome 167 (2019), pp. 34-41.

Résumé

This paper is devoted to the study of the solvability of boundary-value problems for differential equations of the form $$ (\alpha_0(t)+\alpha_1(t)\Delta)u_{tt}-Bu_t-Cu=f(x,t), $$ where $\Delta$ is the Laplace operator acting with respect to spatial variables and $B$ and $C$ are also second-order differential acting with respect to spatial variables. A feature of the equations considered is the condition that the functions $\alpha_0(t)$ and $\alpha_1(t)$ may not possess the fixed sign property on the range $(0,T)$ of the temporal variable; in particular, the operator $\alpha_0(t)+\alpha_1(t)\Delta$ may be irreversible at any point of the interval $(0,T)$, including any strictly inner segments. For problems considered, we prove theorems on the existence and uniqueness of regular solutions (i.e., solutions possessing all generalized derivatives in the Sobolev sense).

Détail

Mots-clés : Sobolev-type equation, existence
Keywords: irreversible operator coefficient, regular solution, uniqueness.

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