Spatially-Nonlocal Boundary Value Problems with the generalized Samarskii--Ionkin condition for quasi-parabolic equations

A. I. Kozhanov ; A. M. Abdrakhmanov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 110-123

Résumé

The work is devoted to the study of the solvability of boundary value problems for quasi-parabolic equations $$(-1)^pD^{2p+1}_tu-\frac{\partial}{\partial x}\left(a(x)u_x\right)+c(x,t)u=f(x ,t)$$ $$((x,t)\in (0,1)\times (0,T), a(x)>0, D^k_t=\frac{\partial^k}{\partial t ^k},\ p>0 - \text{integer})$$ with boundary conditions of one of the types $$u(0,t)-\beta u(1,t)=0, u_x(1,t)=0, t\in (0,T),$$ or $$u_x(0,t)-\beta u_x(1,t)=0, u(1,t)=0, t\in (0,T).$$ The problems under study can be treated as nonlocal problems with the generalized Samarskii–Ionkin condition in terms of spatial variable, for them we prove existence and uniqueness theorems for regular solutions—namely, solutions that have all generalized in the sense of S.L. Sobolev derivatives included in the corresponding equation.

Détail

Mots-clés : quasi-parabolic equations, existence
Keywords: non-local boundary value problems, generalized Samarskii–Ionkin condition, regular solutions, uniqueness.