Existence and regularity of mild solutions in some interpolation spaces for functional partial di erential equations with nonlocal initial conditions

Abstract: This paper is devoted to study the existence and regularity of mild solutions in some interpolation spaces for a class of functional partial di erential equations with nonlocal initial conditions. The linear part is assumed to be a sectorial operator in Banach space X. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. Moreover, we present an example to illustrate the application of main results.


Introduction
Let X be a Banach space with norm ⋅ * and α be a constant such that < α < . We denote by C(J, D(A α )) the Banach space of all the continuous functions from J to D(A α ) provided with the uniform norm topology and D (A α ) the domain of the linear operator A α to be de ned later. In this paper, by using fractional power of operators and Schauder's xed point theorem, we study the existence and regularity of mild solutions in some interpolation to the following functional partial di erential equations with nonlocal initial conditions u ′ (t) + Au(t) = f (t, u(t), u(a (t)), u(a (t)), ⋯, u(a m (t))), t ∈ ( , a], Nonlocal initial conditions can be applied in physics with better e ect than the classical initial condition u( ) = u . For example, in [1] Deng used the nonlocal condition (2) to describe the di usion phenomenon of a small amount of gas in a transparent tube. In this case, condition (2) allows additional measurements at t i , i = , , ⋯, p, which is more precise than the measurement just at t = . In [2], Byszewski pointed out that if c i ≠ , i = , , ⋯, p, then the results can be applied to kinematics to determine the location evolution t → u(t) of a physical object for which we do not know the positions u( ), u(t ), ⋯, u(t p ), but we know that the nonlocal condition (2) holds. Consequently, to describe some physical phenomena, the nonlocal condition can be more useful than the standard initial condition u( ) = u . The importance of nonlocal conditions have also been discussed in [3][4][5][6].
In [7], Fu and Ezzinbi studied the following neutral functional evolution equation with nonlocal conditions x(t)), x(b (t)), ⋯, x(b m (t)))] + Ax(t) = G(t, x(t), x(a (t)), ⋯, x(a n (t))), ≤ t ≤ a, where the operator −A ∶ D(A) ⊂ X → X generates an analytic compact semigroup T(t) (t ≥ ) of uniformly bounded linear operators on a Banach space X, F ∶ [ , a] × X m+ → X, G ∶ [ , a] × X n+ → X, a i , b j , i = , , ⋯, n, j = , , ⋯, m and g are given functions satisfying some assumptions. The authors have proved the existence and regularity of mild solutions. In the subsequent years, various similar results have been established by many authors, see for example [8,9].
Recently, Chang and Liu [10] studied the existence of mild and strong solutions in some interpolation spaces between X and the domain of the linear part for the following semilinear evolution problem with nonlocal initial conditions: where T > , the linear part A is a sectorial operator in X, f and g are given X-valued functions. The aim of this paper is to establish some existence results of (1) and (2) without assuming Lipschitz condition on the nonlinear term and complete continuity on the nonlocal condition. The result obtained is a partial continuation of some results reported in [2,7,[10][11][12]. It is worth mentioning that the theory of fractional power and α-norm are used to discuss the problems so that the results obtained in this chapter can be applied to the systems in which the nonlinear terms involve derivatives of spatial variables, and therefore, they have broader applicability.
The rest of this paper is organized as follows: We introduce some basic de nitions and preliminary facts which will be used throughout this paper in section 2. The existence results of mild solutions are discussed in Section 3 by applying xed point theorem. In Section 4, we provide some su cient conditions to guarantee the regularity of solutions, that is, we obtain the existence of strong solutions. Finally, an example is presented in Section 5 to show the applications of the abstract results obtained.

Preliminaries
Assume A ∶ D(A) ⊂ X → X be a sectorial operator and −A generates an analytic compact semigroup T(t) (t ≥ ) on X. It is easy to see that T(t) (t ≥ ) is exponentially stable, i.e. there exist constants M ≥ and δ < such that is called a growth index of semigroup T(t) (t ≥ ), and at this point v < . For each v ∈ ( , v ), by the de nition of v , there exists a constant M ≥ , such that De ne an equivalent norm in X by then x * ≤ x ≤ M x * . We denoted by T(t) the norm of the operator T(t) in space (X, ⋅ ). By (6), we have and It is well known [13, Chapter 4, Theorem 2.9] that for any u ∈ D(A) and h ∈ C (J, X), the initial value problem of linear evolution equation (LIVP) has a unique classical solution u ∈ C (( , a], X) ∩ C(J, D(A)) expressed by If u ∈ X and h ∈ L (J, X), the function u given by (11) belongs to C(J, X), which is known as a mild solution of the LIVP (10). If a mild solution u of the LIVP (10) belongs to W , (J, X) ∩ L (J, D(A)) and satis es the equation for a.e. t ∈ J, we call it a strong solution.
Throughout this paper, we assume that: Applying (9) and assumption (P ), we get Combining this with the operator spectrum theorem, we know that exists and it is bounded. Furthermore, by Neumann expression, B can be expressed by Therefore, To prove our main results, for any h ∈ C(J, X), we consider the linear evolution equation nonlocal problem Lemma 2.1. If the condition (P ) holds, then the LNP (15)-(16) has a unique mild solution u ∈ C(J, X) given by Proof. By (10) and (11), we know that Eq. (15) has a unique mild solution u ∈ C(J, X) which can be expressed by It follows from (18) that Combining (16) with (19), we have Since By (18) and (21), we know that u satis es (17). Inversely, we can verify directly that the function u ∈ C(J, X) given by (17) is a mild solution of the LNP (15)-(16). This completes the proof.
We recall some concepts and conclusions on the fractional powers of A. Because A ∶ D(A) ⊂ X → X is a sectorial operator, it is possible to de ne the fractional powers A α for < α ≤ . Now we de ne (see [13]) the operator A −α by where Γ denotes the gamma function. The operator A −α is bijective and the operator A α is de ned by We denote by D(A α ) the domain of the operator A α . Furthermore, we have the following properties which appeared in [13].

the linear operator A α T(t) is bounded on X and there exist M α such that
(iv) For every t > , there exists a constant C ′ α such that is a Banach space. We denote it by X α . From now on, for the sake of brevity, we rewrite that (t, u(t), u(a (t)), ⋯, u(a m (t))) ∶= (t, x(t)). (22)

Existence of mild solutions
This section is devoted to the study of the existence of mild solutions for a class of functional partial di erential equations with nonlocal initial conditions (1)- (2). In what follows, we will make the following hypotheses on the data of our problem (1)-(2).
(P )The function f ∶ J × X m+ α → X is Carathéodory continuous and for some positive constant r, there exist constants q ∈ [ , − α), γ > and function ϕ r ∈ L q (J, R + ) such that for any t ∈ J and u j ∈ X α satisfying u j α ≤ r for j = , , , ⋯, m, Theorem 3.1. Assume that the hypotheses (P ) and (P ) are satis ed. Then the problem (1)-(2) has at least one mild solution on C(J, X α ) provided that Proof. We consider the operator Q on C(J, X α ) de ned by With the help of Lemma 2.2, we know that the mild solution of the problem (1)-(2) is equivalent to the xed point of the operator Q de ned by (24). In what follows, we shall prove that the operator Q has at least one xed point by applying the famous Schauder's xed point theorem.
For this purpose, we rst prove that there exists a positive constant R such that the operator Q de ned by (24) maps the bounded closed convex set to D R . If this is not true, there would exist u r ∈ D r and t r ∈ J such that (Qu r )(t r ) α > r for each r > . However, by (9), the condition (P ), Lemma 2.1 and Hölder inequality, we get that . Divided by r on both sides of (25) and then take the lower limits as r → +∞ we get which contradicts with the inequality (23). Therefore, there exists a positive constant R such that the operator Q maps D R to D R .
Below we will verify that Q ∶ D R → D R is a completely continuous operator. From the de nition of operator Q and the assumption (P ) we note that Q is obviously continuous on D R . Next, we shall prove that {Qu ∶ u ∈ D R } is a family of equi-continuous functions. Let u ∈ D R and t ′ , t ′′ ∈ J, t ′ < t ′′ . By (24) one get that It is obvious that Therefore, we only need to check B k α tend to 0 independently of u ∈ D R when t ′′ − t ′ → for k = , , .
For B , by the condition (P ), Lemma 2.1 and Hölder inequality, we have Combining (26) and the strong continuity of the semigroup T(t) (t ≥ ), one can easily get that B α → as t ′′ − t ′ → . For B , taking assumption (P ), Lemma 2.1 and Hölder inequality into account, we obtain For t ′ = , < t ′′ ≤ a, it is easy to see that B = . For t ′ > and < < t ′ small enough, by the condition (P ), Lemma 2.1, Hölder inequality and the equi-continuity of T(t) (t > ), we know that As a result, (Qu)(t ′′ ) − (Qu)(t ′ ) α → independently of u ∈ D R as t ′′ − t ′ → , which means that Q maps D R into a family of equi-continuous functions.
It remains to prove that V(t) = {(Qu)(t) ∶ u ∈ D R } is relatively compact in X α . Obviously it is true in the case t = . Fix t ∈ ( , a], for each ∈ ( , t) and u ∈ D R , de ne

The compactness of T(t) (t > ) ensures that the sets
for every u ∈ D R . Therefore, there are relatively compact sets V (t) arbitrarily close to V(t) for t > . Hence, V(t) is also relatively compact in X α for t ≥ .
Thus, the Ascoli-Arzela theorem guarantees that Q ∶ D R → D R is a completely continuous operator. According to the famous Schauder's xed point theorem we know that the operator Q has at least one xed point u ∈ D R , and this xed point is the desired mild solution of the problem (1)-(2) on C(J, X α ). This completes the proof.
If we replace the condition (P ) by the following condition: (P )The function f ∶ J × X m+ α → X is Carathéodory continuous and there exist a function φ ∈ L q (J, R + ) (q ∈ [ , − α)) and a nondecreasing continuous function ψ ∶ R + → R + such that for all u j ∈ C(J, X α ), j = , , , ⋯, m, and t ∈ J, then we have the following existence result.
Proof. From the proof of Theorem 3.1, we know that the mild solution of the problem (1)-(2) is equivalent to the xed point of the operator Q de ned by (24). In what follows, we prove that there exists a positive constant R such that the operator Q maps the set D R to itself. For any u j ∈ D R , j = , , , ⋯, m, and t ∈ J, by (8), (14), (24), (27), the hypothesis (P ) and Hölder inequality, we have which implies Q(D R ) ⊂ D R . By adopting a completely similar method which used in the proof of Theorem 3.1, we can prove that the problem (1)-(2) has at least one mild solution on C(J, X α ). This completes the proof.
Similarly to Theorem 3.2, we have the following result. (28)

The regularity of solutions
In this section, we discuss the existence of strong solutions for the problem (1)- (2), that is, we shall provide conditions to allow the di erential for mild solutions of the problem (1)- (2). To do this, we need the following lemma: Lemma 4.1 ([12]). If X is a re exive Banach space, then X α is also a re exive Banach space.
Proof. Let Q be the operator de ned in the proof of Theorem 3.1. By the conditions (P ), (P ) and (P ), one can use the same argument as in the proof of Theorem 3.1 to deduce that there exists a constant R > , such that Q(D R ) ⊂ D R . For this R, consider the set for some L * large enough. It is clear that D is a convex, closed and nonempty set. We shall prove that Q has a xed point on D. For any u ∈ D and t ′ , By (8), (30), Lemma 2.1 and the condition (P ), we know that According to the assumptions (P ) and (P ), Lemma 2.1, (29) and (30), we have Using the condition (P ), Lemma 2.1, (30) and (29), we get that Thus, from (30)-(33) we get that where K is a constant independent of L * . Since the condition (P ) implies that Therefore, Q has a xed point u which is a mild solution of the problem (1)-(2). By the above calculation, we see that for this u(⋅) and the following function are Hölder continuous. Since the space X α is re exive by assumption and Lemma 4.1, u(⋅) is almost everywhere di erentiable on ( , a] and u ′ (⋅) ∈ L (J, X). A similar argument shows that F also have this property. Furthermore, we can obtain that Therefore, by (35) we get for almost all t ∈ J that This shows that u is a strong solution of the problem (1)- (2). This completes the proof.

Example
In this section we apply some of the results established in this paper to the following rst order parabolic partial di erential equation with homogeneous Dirichlet boundary condition and nonlocal initial condition where the functions f and a will be described below. Set X = L ([ , π], R) with the norm ⋅ L . Then X is re exive Banach space. De ne an operator A in re exive Banach space X by From [13] we know that −A generates a strong continuous semigroup T(t) (t ≥ ), which is compact, analytic and exponentially stable in X. Furthermore, A has discrete spectrum with eigenvalues x n = n , n ∈ N, associated normalized eigenvectors e n (x) = π sin(nx). Then the following properties hold: To prove the main result of this section, we need the following lemma. Lemma 5.1 ([14]). If u ∈ D(A ), then u is absolutely continuous with u ′ ∈ X and u ′ L = A u L .
According to Lemma 5.1, we can de ne the Banach space X = (D(A ), ⋅ ). Then for u ∈ X , we have We assume that the nonlinear function g satis es the following assumption: (P )The function g ∶ [ , π] × J × R → R is continuous and there is a function h ∈ L ∞ (J, R) such that g(x, t, ζ, ξ, η, ρ) ≤ h(t), for t ∈ J and ζ, ξ, η, ρ ∈ R.
For each t ∈ J and u ∈ X , we de ne a (t))), Then f ∶ [ , ] × X × X → X, and the parabolic partial di erential equation with homogeneous Dirichlet boundary condition and nonlocal initial conditions (36) can be rewritten into the abstract form of problem arctan i = π < , the condition (P ) is satis ed. Below we will verify that f satis es the condition (P ). In fact, it follows from assumption (P ) that In order to obtain the existence of strong solutions to the parabolic partial di erential equation with homogeneous Dirichlet boundary condition and nonlocal initial conditions (36), the following assumptions are also needed.
(P ) The function g ∶ [ , π] × J × R → R is continuous and there is constant L > and α ′ ∈ (α, ) such that (P ) There exist constants l > such that For each φ j , ψ j , ∈ X , j = , and t ′ , t ′′ ∈ J, we have Hence (P ) holds with L = L. Therefore, it from Theorem 4.2, we have the following result.