Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators

The nonlocal boundary value problem for the parabolic differential equation v'(t) + A(t)v(t) = f(t)  (0 ≤ t ≤ T), v(0) = v(λ) + φ, 0 < λ ≤ T in an arbitrary Banach space E with the dependent linear positive operator A(t) is investigated. The well-posedness of this problem is established in Banach spaces C 0 β,γ(E α−β) of all E α−β-valued continuous functions φ(t) on [0, T] satisfying a Hölder condition with a weight (t + τ)γ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.


Introduction: A Cauchy Problem
It is known that (see, e.g., [1][2][3][4][5] and the references given therein) several applied problems in fluid mechanics, physics, and mathematical biology were formulated into nonlocal mathematical models. In general, such nonlocal problems were not well studied.
Before going to discuss the well-posedness of nonlocal boundary value problem, we will give the definition of positive operators in a Banach space and introduce the fractional spaces generated by positive operators that will be needed in the sequel.
Let be a Banach space and let : ( ) ⊂ → be a linear unbounded operator densely defined in . The operator is said to be positive in the Banach space if its spectrum lies in the interior of the sector of angle , 0 < 2 < 2 , symmetric with respect to the real axis, and if, on the edges of this sector, 1 The infimum of all such angles is called the spectral angle of the positive operator and is denoted by ( , ). We call strongly positive in the Banach space if its spectral angle ( , ) < /2. For positive operator in the Banach space , let us introduce the fractional spaces = ( , ) (0 < < 1) consisting of those ] ∈ for which the norm is finite. In the paper [6], the well-posedness in spaces of smooth functions of the nonlocal boundary value problem V ( ) + V ( ) = ( ) (0 ≤ ≤ ) , The Scientific World Journal for the differential equation in an arbitrary Banach space with the strongly positive operator was established. The importance of coercive inequalities (well-posedness) is well known [7,8].
Noted that theory and methods for approximate solutions of local and nonlocal boundary value problems for evolution differential equations have been studied extensively by many researchers (see  and the references therein).
Before going to establish theorems on the well-posedness of nonlocal boundary value problem for parabolic equations in an arbitrary Banach space with the strongly positive dependent space operators, let us consider the abstract Cauchy problem for the differential equation in an arbitrary Banach space with the strongly positive operators ( ) in with domain ( ( )) = , independent of and dense in . A function V( ) is called a solution of the problem (4) if the following conditions are satisfied.
From the existence of the such solutions, it evidently follows that ( ) ∈ ( ) and V 0 ∈ . We say that the problem (4) is well posed in ( ) if the following conditions are satisfied.
An operator-valued function V( , ), defined and strongly continuous jointly in and for 0 ≤ < ≤ , is called a fundamental solution of (4) if (1) the operator V( , ) is strongly continuous in and for 0 ≤ < ≤ , (2) the following identity holds: The Scientific World Journal 3 V( , ) is also called development family, evolution operator, Green's function, and so forth [14]. If the function ( ) is not only continuous, but also continuously differentiable on [0, ], and V 0 ∈ , it is easy to show that the formula gives a solution of problem (4). Now, we will give lemmas and estimates from [14] concerning the semigroup exp{− ( )} ( ≥ 0) and the fundamental solution V( , ) of (4) and theorem on wellposedness of (4) which will be useful in the sequel. Lemma 1. For any 0 < < + < , 0 ≤ ≤ , and 0 ≤ ≤ 1, one has the inequality where does not depend on , , , and .
A function V( ) is said to be a solution of problem (4) in ( ) if it is a solution of this problem in ( ) and the functions V ( ) and ( )V( ) belong to ( ).
As in the case of the space ( ), we say that the problem (4) is well posed in ( ) if the following two conditions are satisfied.
Note that the spaces of smooth functions , 0 ( − ) (0 ≤ ≤ ≤ ,0 < < 1), in which coercive solvability has been established, depend on the parameters , , and . However, the constants in the coercive inequalities depend only on . Hence, we can choose the parameters and freely, which increases the number of functional spaces in which problem (4) is well posed. In particular, Theorems 5 and 6 imply theorems on well-posedness of the nonlocal boundary value problem (4) in ( ) (0 < < 1).
Finally, in the paper [40], the initial-value problem for the fractional parabolic equation in an arbitrary Banach space with the strongly positive operators ( ) was investigated. Here, 1/2 = 1/2 0+ is standard Riemann-Liouville's derivative of order 1/2. The well-posedness of problem (26) in ( ) spaces was established. New exact estimates in Hölder norms for the solution of initial-boundary value problems for fractional parabolic equations were obtained.
In the paper [41], the nonlocal boundary value problem for the parabolic differential equation in an arbitrary Banach space with the strongly positive operators ( ) in with domain ( ( )) = , independent of and dense in , was investigated. The well-posedness of problem (27) in , 0 ( ) spaces was established. New exact estimates in Hölder norms for the solution of three nonlocal boundary value problems for parabolic equations were obtained.
In the present paper, the well-posedness of problem (27) in , 0 ( − ) spaces is established. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.
The paper is organized as follows. Section 1 is introduction. In Section 2, new theorems on well-posedness of problem (27) in , 0 ( − ) spaces are established. In Section 3, theorems on the coercive stability estimates for the solution of two nonlocal boundary value parabolic problems are obtained. Finally, Section 4 is conclusion.

Well-Posedness of Nonlocal Boundary
Value Problem (27) Now, we will give lemmas on the fundamental solution V( , ) of (4) from paper [41].
A function V( ) is called a solution of the problem (27) if the following conditions are satisfied.
The Scientific World Journal 5 A function V( ) is said to be a solution of problem (27) in ( ) if it is a solution of this problem in ( ) and the functions V ( ) and ( )V( ) belong to ( ).
As in the case of the space ( ), we say that the problem (27) is well posed in ( ) if the following two conditions are satisfied.
Furthermore, the method of proof of Theorem 6 and scheme of proof of Theorem 9 enable us to establish the following theorem on well-posedness of (27) in spaces , 0 ( − ), 0 ≤ ≤ ≤ , 0 < < 1.
Note that the spaces of smooth functions , 0 ( − ) (0 ≤ ≤ ≤ , 0 < < 1), in which coercive solvability has been established, depend on the parameters , , and . However, the constants in the coercive inequalities depend only on . Hence, we can choose the parameters and freely, which increases the number of functional spaces in which problem (27) is well posed. In particular, Theorems 9 and 10 imply theorems on well-posedness of the nonlocal boundary value problem (27) in , 0 ( )(0 ≤ ≤ , 0 < < 1) which is established in the paper [41].
We introduce the Banach spaces [0, 1] (0 < < 1) of all continuous functions ( ) satisfying a Hölder condition for which the following norms are finite: It is known that the differential expression The proof of Theorem 11 is based on abstract Theorem 9 and on the following theorem on the structure of fractional spaces ( [0, 1], , ).