On the well-posedness of the nonlocal boundary value problem for elliptic-parabolic equations

The abstract nonlocal boundary value problem 8 − d 2 u(t) dt2 + sign(t)Au(t) = g(t),(0 � t � 1), du(t) dt + sign(t)Au(t) = f(t),(−1 � t � 0), u(1) = u(−1) + µ for the differential equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. The well-posedness of this problem in Holder spaces without a weight is established. The coercivity inequalities for solutions of the boundary value problem for elliptic-parabolic equations are obtained.

First of all, let us give some estimates that will be needed below.
Lemma 1.2 [37].For 0 < α < 1 the norms of the spaces E α (A Lemma 1.3 .For 0 < α < 1 the following estimates hold: e −A || H→Eα(A where C([a, b], H) stands for the Banach space of all continuous functions ϕ(t) defined on [a, b] with values in H equipped with the norm Then the following estimates hold: where M does not depend on α, f (t) and g(t).
Proof.Using estimates (1.2)-(1.3),we get Ae −(s+z)A for all z, z > 0 and g(t) ∈ C α ([0, 1], H).Using estimates (1.2)-(1.3),we get for all z, z > 0 and f (t 3), we get for all z, z > 0 and g(t for all z, z > 0 and all s, s > 0, we have the bounded Then for all z, z > 0 and g(t 3), we get A solution of problem (1.1) defined in this manner will from now on be referred to as a solution of problem (1.1) in the space We say that the problem (1.1) is well-posed in C(H), if there exists the unique solution and the following coercivity inequality is satisfied: where M does not depend on µ, f (t) and g(t).
In fact, inequality (1.23) does not, generally speaking, hold in an arbitrary Hilbert space H and for the general unbounded self-adjoint positive definite operator A. Therefore, the problem (1.1) is not well-posed in C(H) [8].The well-posedness of the boundary value problem ( As in the case of the space C(H), we say that the problem (1.1) is well-posed in F (H), if the following coercivity inequality is satisfied: where M does not depend on µ, f (t) and g(t).
In paper [41] the well-posedness of problem (1.1) in Hölder spaces C α,α ([−1, 1], H), (0 < α < 1) with a weight was established.The coercivity inequalities for the solution of boundary value problems for elliptic-parabolic equations were obtained.The first order of accuracy difference scheme for the approximate solution of the nonlocal boundary value problem (1.1) was presented.The well-posedness of this difference scheme in Hölder spaces with a weight was established.In applications, the coercivity inequalities for the solution of difference scheme for elliptic-parabolic equations were obtained.
Then the boundary value problem (1.1) is well-posed in a Holder space C α (H) and the following coercivity inequality holds: where M does not depend on α, f (t), g(t) and µ.
Since the operator has an inverse it follows that Second, we will establish estimate (1.25).It is based on the estimates for the solution of an inverse Cauchy problem (1.27) and on the estimates Eα(A for the solution of the boundary value problem (1.26) and on the estimates EJQTDE, 2011 No. 49, p. 8 Au 1 − g( 1) Eα(A for the solution of the boundary value problem (1.1).Estimates (1.33) and (1.34) were established in [9] and [10].Now, first step would be to establish (1.35).Using (1.32), we get Using this formula and estimates ( , we obtain ) ds EJQTDE, 2011 No. 49, p. 9 Second step would be to establish (1.36).Using (1.32), we get (f (0) + g(0)) .
First, the mixed boundary value problem for the elliptic-parabolic equation generated by the investigation of the motion of gas on the nonhomogeneous space is considered (see [6] and [40]).Problem (2.1) has a unique smooth solution u(t, x) for the smooth a(x) a > 0(x ∈ (0, 1)), and  ]) .Here M does not depend on α, f (t, x) and g(t, x).
The proof of Theorem 2.1 is based on the abstract Theorem 1.5 and the symmetry properties of the space operator generated by the problem (2.1).
The proof Theorem 2.2 is based on the abstract Theorem 1.5 and the symmetry properties of the space operator A generated by the problem (2.2) and the following theorem on the coercivity inequality for the solution of the elliptic differential problem in L 2 (Ω). is valid.
]) functions and δ = const > 0. This allows us to reduce the mixed problem(2.1)to the nonlocal boundary value problem (1.1) in a Hilbert space H = L 2 [0, 1] with a self-adjoint positive definite operator A defined by (2.1).Theorem 2.1 .The solutions of the nonlocal boundary value problem (2.1) satisfy the coercivity inequality

Theorem 2 . 3 .
For the solutions of the elliptic differential problem n r=1 (a r (x)u xr ) xr = ω(x), x ∈ Ω, (2.3) u(x) = 0, x ∈ S the following coercivity inequality [36] n r=1 u xr xr L2(Ω) ≤ M ||ω|| L2(Ω) 1.1) can be established if one considers this problem in certain spaces F (H) of smooth H-valued functions on [−1, 1].A function u(t) is said to be a solution of problem (1.1) in F (H) if it is a solution of this problem in C(H) and the functions u ′′