NONLOCAL FINAL VALUE PROBLEM GOVERNED BY SEMILINEAR ANOMALOUS DIFFUSION EQUATIONS

. Our goal is to establish some suﬃcient conditions for the solvability of the nonlocal ﬁnal value problem involving a class of partial diﬀerential equations, which describes the anomalous diﬀusion phenomenon. Our analysis is based on the theory of completely positive functions, resolvent operators and ﬁxed point arguments in suitable function spaces. Especially, utilizing the regularity of resolvent operators, we are able to deal with non-Lipschitz cases. The obtained results, in particular, extend recent ones proved for fractional diﬀusion equations.


Introduction.
Let Ω ⊂ R d be a bounded domain with smooth boundary ∂Ω. Consider the following equation with the nonlocal final condition u(T, ·) = g(u) in Ω, (2) and the Dirichlet boundary condition u = 0 on ∂Ω, t ≥ 0.
Equation (1) belongs to a class of nonlocal partial differential equations, which has been employed to describe anomalous diffusion processes, as remarked in [14].
In order to deal with (1), we use the following standing hypothesis. (PC) The kernel function k ∈ L 1 loc (R + ) is nonnegative and nonincreasing, and there exists a function l ∈ L 1 loc (R + ) such that k * l = 1 on (0, ∞).
As far as the problem (1)-(3) is concerned, the objective is to detect the previous state of process from its present one, in the situation that, the process cannot be observed at the time t = 0 and then the initial data is not known. When the measured final value depends on some information, e.g. the energy, of the system, the problem involves a nonlocal final condition like (2). Unlike the problem with nonlocal initial condition (i.e. u(0) = g(u), forward problem), the nonlocal final value problem is of backward type and much more complicated. The main reason is the smoothing effect of the forward problem, i.e. u(t), for t > 0, is more regular than u(0). Consequently, t = 0 may become a singular point of u if the given final value is not regular enough (see Remark 2).
In the case k(t) = g 1−α (t) (fractional/slow diffusion case), there has been a number of works devoted to (1)- (3). The linear problem (f and g are independent of u) was considered in [10,11,17,18,19], where the problem was proved to be ill-posed in the sense that, the solution is unstable with respect to final data, and some regularization methods were proposed to obtain the solution. The semilinear problem with final value condition (g is independent of u) was addressed in [13]. Recently, the semilinear problem with nonlocal final value condition was solved in [12]. It should be mentioned that, the analysis in these works is based on the formulation of solution utilizing the Mittag-Leffler functions. Taking other cases into account, for instant, ultra-slow diffusion or multi-term fractional diffusion case, such a nice representation is unavailable. The aim of our work is to give a comprehensive analysis for solving the problem in a general case, i.e. when k is a Sonine kernel function. From the technical point of view, our approach employs the theory of completely positive functions (see [2]) and resolvent operators (see, e.g. [9]), combining with fixed point principles. Particularly, using the parabolicity of the anomalous diffusion equation, we get a regularity of resolvent operators, which enable us to solve the problem in non-Lipschitz cases. In comparison with the results made for the fractional diffusion case, we deal with a more general class of nonlinearity functions and give more concrete conditions. Our paper is organized as follows. In the next section, we give a presentation of mild solution to (1)-(3) in linear case using resolvent operators. Section 3 is devoted to the existence results under regular setting, that is, f is defined on [0, T ] × L 2 (Ω) and g is defined on C([0, T ]; L 2 (Ω)). With this setting, we obtain mild solutions in C([0, T ]; L 2 (Ω)). In Section 4, we prove the solvability of (1)-(3) in the space C l ((0, T ]; L 2 (Ω)), consisting of functions possibly discontinuous at t = 0. In the last section, we show two examples demonstrating the obtained results.
2. Preliminaries. Consider the following scalar integral equations The existence and uniqueness of s and r were analyzed in [6]. Recall that the function l is called a completely positive kernel iff s(·) and r(·) take nonnegative values for every µ > 0. The complete positivity of l is equivalent to that (see [1]), there exist α ≥ 0 and k ∈ L 1 loc (R + ) nonnegative and nonincreasing which satisfy αl + l * k = 1. So the kernel function l taken from the hypothesis (PC) is completely positive. In the case l(t) = g α (t), by using the Laplace transform, one can see that is the Mittag-Leffler function. Denote by s(·, µ) and r(·, µ) the solutions of (4) and (5), respectively. As mentioned in [15], the functions s(·, µ) and r(·, µ) take nonnegative values for all µ ∈ R. Some additional properties of these functions are collected in the following proposition. Proposition 1. Let the hypothesis (PC) hold. Then for every µ > 0, s(·, µ), r(·, µ) ∈ L 1 loc (R + ). In addition, we have: (a) The function s(·, µ) is nonincreasing. Moreover, , ∀t ≥ 0.
Let {e n } be the orthonormal basis of L 2 (Ω) consisting of eigenfunctions of the operator −∆ with homogeneous Dirichlet boundary condition, i.e., −∆e n = λ n e n in Ω, e n = 0 on ∂Ω, where we can assume that {λ n } is nondecreasing with λ 1 > 0. Then the operator of fractional power can be defined by for β ∈ R, here the notation (·, ·) denotes the inner product in L 2 (Ω). Let V β = D((−∆) β ), then V β is a Hilbert space with the inner product Furthermore, for any β > 0, the embedding V β ⊂ L 2 (Ω) is compact. We now define the following operators It is easily seen that S(t) and R(t) are linear. We show some basic properties of these operators in the following lemma.
Remark 1. Since the embedding V γ ⊂ L 2 (Ω) is compact, it follows from (11) that, for any t > 0, S(t) : then (13) ensures that Q t is also compact as an operator from C([0, T ]; L 2 (Ω)) into L 2 (Ω), by reasoning that the embedding V γ 2 ⊂ L 2 (Ω) is compact. In what follows, we use the notation u(t) for u(t, ·) and consider u as a function defined on [0, T ], taking values in space Let Then (15) is formally reduced to the relaxation equation The last equation can be rewritten as which implies that thanks to Proposition 1(e). Hence The final condition u(T ) = g leads to which yields So we have the following representation for the solution of (15)- (16). Let Proposition 2. We have the following estimates: (a) For v ∈ L 2 (Ω) and t ∈ (0, T ], Proof. (a) In view of Proposition 1(a), one gets For 0 < t ≤ T , it follows from the relation k * l = 1 that due to the fact that k is nonincreasing. Then Let v ∈ V γ . Then which ensures (20).
The proof is complete.
The following result shows the condition for problem (15)- (16) to have a unique solution.
Proof. The proof is similar to that in [11] and we omit it.
In the next sections, we will analyze sufficient conditions for the solvability of (1)-(3).
3. Solvability in regular setting. We first consider the simple case, where f and g are Lipschitzian and take values in V γ .
In the next theorem, we show that if the kernel function l is nonincreasing, then f can take values in L 2 (Ω). where Then the problem (1) Proof. Since l is nonincreasing, we have , ∀t > 0, thanks to (6). Then Consider the solution operator defined by (24). Let u 1 , u 2 ∈ C([0, T ]; L 2 (Ω)). We get as in (25). In addition, one has Finally, we observe that, for all t ∈ [0, T ], here we utilized (6), (30), and the estimates , ∀n. Therefore, It follows from (31)-(33) that which shows that Φ is a contraction mapping. The proof is complete.
Remark 2. 1. It was shown in [1] that, if k is completely monotone, then the associate kernel function l is nonincreasing.
2. Let us discuss about condition (29). If L f is a nonnegative function satisfying that L f (T ) = 0 and that then condition (29) is testified. Indeed, we see that Then which guarantees (29).
3. If the function g takes "less regular" values, e.g. g(u) ∈ V β with β < γ, the problem (1)-(3) may fail to admit solution in C([0, T ]; L 2 (Ω)). Indeed, we consider a simple case, when g is independent of u. Assume the opposite, that the problem has a solution u ∈ C([0, T ]; L 2 (Ω)), i.e. u(0) is well-defined. Then Let the assumptions (28)-(29) hold and f (τ, 0) = 0. Then the last two terms are defined at t = 0. Regarding the first term, let Then g ∈ V β and Note that λ n ∼ Cn 2 d as n → ∞, for some C > 0, then which implies that P (0)g = ∞. This is a contradiction.
We are now concerned with the case when f and g are non-Lipschitzian. In order to deal with the problem in this case, we need a regularity of the operators S(·) and R(·). Recall that S(t), defined by (8), is the resolvent operator of the problem that is, u(t) = S(t)u 0 . This problem is equivalent to the Volterra equation with A = (−∆) γ . This can be seen by convoluting (34) with l.
In what follows, for l ∈ L 1 loc (R + ), we denote byl the Laplace transform of l. We recall some notions and facts stated in [9].  loc (R + ) is of subexponential growth and θ-sectorial for some θ < π. If A is closed linear densely defined, such that ρ(−A) ⊃ Σ(0, θ), and then equation (36) is parabolic.
The following result on the regularity of resolvent operator for equation (36) will be used in the sequel. Proposition 5. [9, Theorem 3.1] Assume that (36) is parabolic and the kernel function l is m-regular for some m ≥ 1. Then there is a resolvent family S(·) ∈ C (m−1) ((0, ∞); L(L 2 (Ω))) for (36), and a constant M ≥ 1 such that here L(L 2 (Ω)) denotes the space of bounded linear operators on L 2 (Ω).
By this proposition, we are able to obtain a regularity of the operator R(·).
Proof. We first show that Φ 1 is a compact operator. Observe that, for v ∈ V 2γ , we have here we used estimate (21). Let D ⊂ C([0, T ]; L 2 (Ω)) be a bounded set, i.e. there is ρ > 0 such that u ∞ ≤ ρ, ∀u ∈ D. Then by (G2), g(D) is bounded in V 2γ . According to (41), P (t)g(D) is bounded in V γ , which implies that the set P (t)g(D) is compact in L 2 (Ω) for any t ∈ [0, T ]. Similarly, let then F(D) is bounded in V 2γ , thanks to (F2). Thus the set P (t)F(D) is compact in L 2 (Ω) as well. We have proved that Φ 1 (D)(t) is compact in L 2 (Ω) for any t ∈ [0, T ]. We now testify the equicontinuity of Φ 1 (D) = P (·)[g(D) + F(D)]. In view of (F2)-(G2), one can take R > 0 such that g(D) here we used Proposition 1(b) and the estimate 1 λ γ n s(T, λ γ n ) is an equicontinuous set and then it is relatively compact in C([0, T ]; L 2 (Ω)), according to the Arzela-Ascoli theorem. In order to show the relative compactness of Φ 2 (D), one observes that Regarding E 1 , we get Dealing with E 2 , one can assume that t > 0 and 0 < √ h < t. Then Using the same estimates as those for E 1 , we have For the last integral E 2c , we see that uniformly in u ∈ D, here we employed the regularity of R(·) obtained in Lemma 3.5.
Proof. We show that the solution operator admits a fixed point in C([0, T ]; L 2 (Ω)) by using the Schauder fixed point theorem. By Lemma 3.6, we see that Φ is a compact mapping. It suffices to prove that, there exists ρ > 0 such that Φ(B ρ ) ⊂ B ρ , where B ρ is the closed ball in C([0, T ]; L 2 (Ω)) with radius ρ and center at origin. Assume to the contrary that, for each n ∈ N, there is u n ∈ C([0, T ]; L 2 (Ω)) such that u n ∞ ≤ n and Φ(u n ) ∞ > n. Then we get thanks to Proposition 2(b) and Lemma 2.1(b). Using the estimates v ≤ λ −2γ , and the assumptions (F2)-(G2), we obtain It follows that Passing to the limit as n → ∞ in the last relation, we get a contradiction, due to condition (42). The proof completes.
In this section, we consider the case that f and g take less regular values than those assumed in Section 3, and find a solution of the problem (1)-(3) in C l ((0, T ]; L 2 (Ω)).
which shows the inclusion.
Passing to the limit as n → ∞, one gets a contradiction with (50). The proof is complete.
where H and K are given functions. In this case, V γ = V = H 2 (0, 1) ∩ H 1 0 (0, 1). Note that the associate kernel function l(t) = ∞ 0 e −pt 1 + p dp is nonincreasing on thanks to the fact that, the function s → ln(1 + s 2 ) is Lipschitzian with constant 1.
We have proved that, the kernel function l is 3-regular.
Concerning the function g, applying the Hölder inequality yields where L g = T  Taking estimate (57) into account, we get g(u) 2 V ≤ ( g 2 + g 2 )L 2 g u 2δ C l .