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

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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).
Mots-clés : Sobolev-type equation, existence
Keywords: irreversible operator coefficient, regular solution, uniqueness. 
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     author = {A. I. Kozhanov},
     title = {Boundary value problems for {Sobolev} type equations with irreversible operator coefficient of the highest derivatives},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {34--41},
     publisher = {mathdoc},
     volume = {167},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_167_a4/}
}
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A. I. Kozhanov. Boundary value problems for Sobolev type equations with irreversible operator coefficient of the highest derivatives. 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. http://geodesic.mathdoc.fr/item/INTO_2019_167_a4/