Note on abstract elliptic equations with nonlocal boundary in time condition

Our main purpose of this paper is to study the linear elliptic equation with nonlocal in time condition. The problem is taken in abstract Hilbert space H. In concrete form, the elliptic equation has been extensively investigated in many practical areas, such as geophysics, plasma physics, bioelectric field problems. Under some assumptions of the input data, we obtain the well-posed result for the solution. In the first part, we study the regularity of the solution. In the second part, we investigate the asymptotic behaviour when some paramteres tend to zero.


Introduction
Let H be a Hilbert space. Let L : D(L) ⊂ H → H be a positive-definite, self-adjoint operator with compact inverse on H. Let us assume that A admits an orthonormal eigenbasis {ϕ k } k≥1 in H, associated with the eigenvalues of the operator L and 0 < λ 1 ≤ λ 2 ≤ · · ·λ j ≤ ..., and lim j→∞ λ j = ∞. Let T > 0 be a given real number. In this paper, we consider the nonlinear elliptic equation          ∂ 2 u ∂y 2 = Lu + F (y), y ∈ (0, T ), u y (0) = 0, ∈ (0, T ), where f and F called input data and defined later. The problem (1.1) may called abstract elliptic equations with nonlocal boundary in time condition. Non-local boundary value issues are undoubtedly one of the areas that excel in many different fields of application, such as chaos, chemistry, biology, and physics. In problem (1.1), if α = 0, = 1 then we called the Cauchy problem for elliptic problem which also has been studied in many paper, for example [13,14,18,19,20,21]. In the abstract framework of operators on Hilbert spaces, regularization techniques are developed by B. Kaltenbacher et al [15,16,17] .
To the best of our knowledge, there are not any paper concern to Problem (1.1). Our work is probably one of the first results on this type of problem for elliptic equations. Our contribution for this paper are described as follows • The first contribution is the investigation of the solution space and the regularity of the solutions.
• The second contribution is to demonstrate the convergence of solutions when the parameters reach zero.

Nonlocal in time elliptic equation
For positive number r ≥ 0, we also define the Hilber scale space . Let us also define the space of Geverey type V s,T be as follows for s ∈ R, T > 0. The associated norm on V s,T is given by Theorem 2.1. Let u * be the solution of Problem (1.1) with the case α = 1, = 0. Let f be the function belongs to D(A ν− θ 2 ) and F ∈ L ∞ (0, T ; V ν−θ/2−1/2,T ) for any 0 < θ < 1 and ν > 0. Then we get u * ∈ L 1 (0, T ; D(A ν )) and the following estimate holds Proof. The mild solution of Problem (1.1) in the case of α = 1, = 0 is given by For the term I 1 , using the inequality we get the following estimate . (2.7) Hence, we obtain . (2.8) The second term I 2 is bounded by Noting that for y ∈ [0, T ], we get we get that Therefore, we obtain that From the inequality (2.6), the third term I 3 is estimated as follows Using (2.10), we find that which allows us to get that Combining (2.5), (2.8), (2.12) and (2.15), we arrive at Hence, due to the proper integral T 0 (T − y) −θ dy is convergent, we can deduce that u * ∈ L 1 (0, T ; D(A ν )) and the following estimate holds Theorem 2.2. Let f and F be as Theorem (2.1). Let u α, be the solution of ... Moreover, we have lim →0 u 1, = u * and the following convergent is true Proof. We divide the proof into two parts. Part 1. Existence and regularity of u 1, . Let us assume that u(0) = u 0 ∈ H. Then we have the expression of u as in Fourier series u(y) = ∞ j=1 u(y), ψ j ψ j , where u(y), ψ j is Fourier coefficient of u. Thanks to the work of [8], the Fourier coefficient of u satisfies that the following equality This implies that This implies that By the properties of Fourier series, the mild solution to Problem (1.1) is given by Using (2.6), we get the following estimate we get the following estimate .

(2.26)
From the inequality (2.6), the third term J 3 is estimated as follows Therefore, we can deduce that (2.28) Part 2. The convergence of u 1, and u * when → 0. When α = 1, we have the following fomula Since the representations of u 1, and u * , we find that By a simple caculation, we obtain Now, we focus on the first term K 1 . Using the inequality By looking at the inequality e −z ≤ C θ z −θ , we obtain the following estimate This implies that . (2.34) Hence, we derive the following estimate .
(2.35) Next, we continue to treat the second term K 2 (y). Using (2.33) and Hölder inequality, it it easy to observe that