ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 7, pp. 841–854.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
A.N. Artyushin, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 867–881.
PARTIAL DIFFERENTIAL EQUATIONS
Volterra Integral Equations and Evolution Equations
with an Integral Condition
A. N. Artyushin
Data East, LLC, Novosibirsk, 630090 Russia
Received September 14, 2016
Abstract—We suggest a simple method for reducing problems with an integral condition for
evolution equations to a Volterra integral equation of the ﬁrst kind. For Volterra equations of
the convolution type, we indicate necessary and suﬃcient solvability conditions for the case in
which the right-hand side lies in some classes of functions of ﬁnite smoothness. We use these
conditions to construct examples of nonexistence of a local solution for the heat equation with
an integral condition.
Given a function K(x) ∈ L
(0,π), consider the following problem in the rectangle
Q =(0,π) × (0,T):
(x, t) − u
(x, t)=0, (1.1)
u(x, 0) = 0,x∈ (0,π), (1.2)
u(0,t)=0,t∈ (0,T), (1.3)
K(x)u(x, t) dx = ϕ(t),t∈ (0,T). (1.4)
u(x, t) ∈ L
is called a generalized solution of problem (1.1)–(1.4) if it satisﬁes condition (1.4) for almost all
t ∈ (0,T)andtherelation
(x, t)) dx dt =0
holds for each smooth function ψ(x, t) such that
ψ(x, T )=0,x∈ (0,π),
One typical approach to the solution of this problem is as follows. One multiplies Eq. (1.1) by the
function K(x) and integrates by parts, thus obtaining some nonlocal boundary conditions instead
of the integral condition (1.4). This method was used in  to reduce the problem to a Volterra
integral equation of the ﬁrst kind with the use of the fundamental solution of the heat equation.