Пространственно-нелокальные краевые задачи с обобщенным условием Самарского-Ионкина для квазипараболических уравнений

Abstract

 Abstract. The work is devoted to the study of the solvability of boundary value problems for quasi-parabolic equations
$$(-1)^pD^{2p+1}_tu-\frac{\partial}{\partial x}\left(a(x)u_x\right)+c(x,t)u=f(x ,t)$$
$$((x,t)\in (0,1)\times (0,T),\quad a(x)>0,\quad D^k_t=\frac{\partial^k}{\partial t ^k},\quad p>0\mbox{\it --- integer})$$
with boundary conditions of one of the types
$$u(0,t)-\beta u(1,t)=0,\quad u_x(1,t)=0,\quad t\in (0,T),$$
or
$$u_x(0,t)-\beta u_x(1,t)=0,\quad u(1,t)=0,\quad t\in (0,T).$$
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.