ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 7, pp. 879–890.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
A.I. Kozhanov, G.A. Lukina, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 906–918.
PARTIAL DIFFERENTIAL EQUATIONS
Spatially Nonlocal Problems with Integral Conditions
for Third-Order Diﬀerential Equations
A. I. Kozhanov
and G. A. Lukina
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences,
Novosibirsk, 630090 Russia
Novosibirsk State University, Novosibirsk, 630090 Russia
Polytechnic Institute (Branch) of North-Eastern Federal University, Mirny, 678174 Russia
Received September 14, 2016
Abstract—We obtain suﬃcient conditions for the existence of regular solutions of some non-
local problems for the equation u
+ μu = f(x, t) with conditions containing integrals
with respect to the spatial variable.
There are quite a few papers, including recent ones, that deal with problems involving conditions
of integral type with respect to the spatial variables. In particular, such problems were studied
in [1–9] for parabolic equations, in [10–22] for hyperbolic equations, and in [22–26] for some nonclas-
sical equations such as pseudo-parabolic and pseudo-hyperbolic ones. At the same time, problems
with conditions of integral type with respect to the spatial variables for odd-order equations
+ μu = f(x, t),
which are sometimes called elliptic–parabolic or (2m +1)-parabolic equations [27, 28], are little
studied; one can only indicate the papers [29–31]. This gap is partly bridged in the present paper.
Note that, when studying the solvability of nonlocal problems with integral conditions, one often
uses a trick (based on integration by parts) that reduces a problem with purely integral conditions to
a problem with “semi-integral” conditions relating the values of the solution and/or its derivatives
at the boundary points to the values of some integrals of the solution (e.g., see ). We do not
use this trick in the present paper. (In other word, the kernels of integral conditions are allowed to
have high-order zeros at the boundary points.)
The present paper deals with a model case of the equation given above, namely, with the case
of m =1. We discuss more general cases as well as possible generalizations at the end of the paper.
The approach used in the present paper is close to that in , where spatially nonlocal problems
were studied for some classes of nonstationary equations, classical as well as nonclassical.
1. STATEMENT OF THE PROBLEMS
Let Ω be the interval (0, 1) of the axis Ox,letQ be the rectangle Ω × (0,T), 0 <T <+∞,
let K(x),N(x), and f(x, t) be given functions deﬁned for x ∈
Ωand(x, t) ∈ Q, respectively, and let
μ be a given real number. Further, let L, l
, and l
be the operators whose action on a function
v(x, t) is deﬁned by the formulas
Lv = v
,i=1,...,4, are given real numbers.