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

1 Introduction

Let X be a Banach space with norm |·|^∗ and α be a constant such that 0 < α< 1. 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 defined later.

In this paper, by using fractional power of operators and Schauder’s fixed point theorem, we study the existence and regularity of mild solutions in some interpolation to the following functional partial differential equations with nonlocal initial conditions

where u(·) takes values in a subspace spaces of Banach space X, A : D(A) ⊂ X → X is a sectorial operator, m is a positive integer number, J = [0, a], a > 0 is a constant, a_j : J → J are continuous functions such that 0 ≤ a_j(t) ≤ t for j = 1, 2, …, m, f : J × (D(A^α))^m+1 → X is Carathéodory continuous nonlinear function, 0 < t₁ < t₂ < … < t_p < a, c_i are real numbers, c_i ≠ 0, i = 1, 2, …, p.

Nonlocal initial conditions can be applied in physics with better effect than the classical initial condition u(0) = u₀. For example, in [1] Deng used the nonlocal condition (2) to describe the diffusion phenomenon of a small amount of gas in a transparent tube. In this case, condition (2) allows additional measurements at t_i, i = 1, 2, …, p, which is more precise than the measurement just at t = 0. In [2], Byszewski pointed out that if c_i ≠ 0, i = 1, 2, …, 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(0),u(t₁), …, 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(0) = u₀. 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

where the operator –A : D(A) ⊂ X → X generates an analytic compact semigroup T(t) (t ≥ 0) of uniformly bounded linear operators on a Banach space X, f : [0, a] × X^m+1 → X, G : [0, a] × Xⁿ⁺¹ → X, a_i, b_j, i = 1, 2, …, n, j = 1, 2, …, 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 > 0, 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 definitions 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 fixed point theorem. In Section 4, we provide some sufficient 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.

2 Preliminaries

Assume A : D(A) ⊂ X → X be a sectorial operator and –A generates an analytic compact semigroup T(t) (t ≥ 0) on X. It is easy to see that T(t) (t ≥ 0) is exponentially stable, i.e. there exist constants M ≥ 1 and δ < 0 such that

the infimum of δ

is called a growth index of semigroup T(t) (t ≥ 0), and at this point v₀ < 0. For each v ∈ (0,|v₀|), by the definition of v₀, there exists a constant M ≥ 1, such that

Define 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 u₀ ∈ D(A) and h ∈ C¹(J, X), the initial value problem of linear evolution equation (LIVP)

has a unique classical solution u ∈ C¹((0, a], X) ∩ C(J, D(A)) expressed by

If u₀ ∈ X and h ∈ L¹(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^{1, 1}(J, X) ∩ L¹(J, D(A)) and satisfies the equation for a.e. t ∈ J, we call it a strong solution.

Throughout this paper, we assume that: (P0)∑i=1p|ci|<1.

Applying (9) and assumption (P₀), we get ∥∑i=1∞ciT(ti)∥≤∑i=1p|ci|e−vt1<1. 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 (LNP)

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 define the fractional powers A^α for 0 < α ≤ 1. Now we define (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 defined by

We denote by D(A^α) the domain of the operator A^α.

Furthermore, we have the following properties which appeared in [13].

D(A^α) endowed with the norm ∥x∥_α = ∥A^αx∥ for all x ∈ D(A^α) is a Banach space. We denote it by X_α.

From now on, for the sake of brevity, we rewrite that

3 Existence of mild solutions

This section is devoted to the study of the existence of mild solutions for a class of functional partial differential equations with nonlocal initial conditions (1)-(2). In what follows, we will make the following hypotheses on the data of our problem (1)-(2).

1. The function f:J×Xαm+1→X is Carathéodory continuous and for some positive constant r, there exist constants q ∈ [0, 1 – α), γ > 0 and function φr∈L1q(J,R+) such that for any t ∈ J and u_j ∈ X_α satisfying ∥u_j∥_α ≤ r for j = 0, 1, 2, …, m,

   ∥f(t,u0,u1,⋯,um)∥≤φr(t),lim infr→+∞∥φr∥L1q([0,a])r:=γ<+∞.

If we replace the condition (P₁) by the following condition:

1. The function f:J×Xαm+1→X is Carathéodory continuous and there exist a function ϕ∈L1q(J,R+) (q ∈ [0, 1 – α)) and a nondecreasing continuous function ψ : ℝ⁺ → ℝ⁺ such that

   ∥f(t,u0,u1,⋯,um)∥≤ϕ(t)ψ(∑j=0m∥uj∥α),

   for all u_j ∈ C(J, X_α), j = 0, 1, 2, …, m, and t ∈ J,

then we have the following existence result.

Similarly to Theorem 3.2, we have the following result.

4 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 differential for mild solutions of the problem (1)-(2). To do this, we need the following lemma: