ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 7, pp. 864–878.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
A.V. Glushak, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 891–905.
PARTIAL DIFFERENTIAL EQUATIONS
Abstract Cauchy Problem
for the Bessel–Struve Equation
A. V. Glushak
Belgorod National Research University, Belgorod, 308015 Russia
Received November 8, 2016
Abstract—We consider the Cauchy problem for the Bessel–Struve equation in a Banach space.
A suﬃcient condition for the solvability of this problem is proved, and the solution operator is
written in explicit form via the Bessel and Struve operator functions. A number of properties
is established for the solutions.
Let A beaclosedoperatorwithdensedomainD(A)inaBanachspaceE.Fork>0, consider
the Euler–Poisson–Darboux equation
It follows from the results in [1, 2] that well-posed initial conditions for the Euler–Poisson–
Darboux equation (1) at the point t =0havetheform
u(0) = u
(0) = 0, (2)
the initial condition u
(0) = 0 being dropped for k ≥ 1, which is typical of many equations with
a singularity in the coeﬃcients at t =0.
A well-posed statement of the initial conditions depending on the parameter k ∈
, as well as
the solution of the corresponding Cauchy problems for the case in which A is the Laplace operator
with respect to the spatial variables, is given in the monograph [3, Ch. 1]. Subsequent results on
the theory of singular partial diﬀerential equations can be found in [4–6]. For the terminology used
here, see the survey papers [7, 8]. The abstract Euler–Poisson–Darboux equation (1) was studied
in [9; 10, Ch. 1; 11] under various assumptions about the operator A.
The set of operators A for which problem (1), (2) is uniformly well posed will be denoted by G
and the solution operator of this problem will be denoted by Y
(t) and called the Bessel operator
function. Problem (1), (2) with k = 0 is uniformly well posed only if the operator A is the generator
of a cosine operator function C(t).
The papers [1, 2] give conditions on A ensuring the well-posed solvability of problem (1), (2).
They are stated in terms of an estimate for the norm of the resolvent R(λ, A)anditsweighted
derivatives in  and in terms of a fractional power of the resolvent and its ordinary derivatives
For 0 <k<1, there exists a more general well-posed statement of the initial conditions.
Consider initial conditions of the form
u(0) = u
The Bessel operator function Y
(t) can also be used for the solution of the weighted Cauchy prob-
lem (1), (3) for the Euler–Poisson–Darboux equation. If u
∈ D(A)andA ∈ G
the unique solution of the Cauchy problem (1), (3) has the form 
1 − k