Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 64, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.64 EXISTENCE AND UNIQUENESS OF THE SOLUTION TO INITIAL AND INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS WITH FRACTIONAL LOAD RAVI P. AGARWAL, UMIDA BALTAEVA, FLORENCE HUBERT, BOBURJON KHASANOV Abstract. This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. 1. Introduction The theory of differential equations with fractional operators has been widely used in various fields of science and engineering [46]. These equations are mul- tidisciplinary and used in diverse fields such as dynamical systems, control the- ory, elasticity, electric drives, circuits systems, continuum mechanics, heat transfer, quantum mechanics, fluid mechanics, signal analysis, biomathematics, biomedicine, social systems, and bioengineering for modeling of the anomalous diffusion processes [2, 32, 47]; see the references therein. Fractional diffusion equations are extensions of the basic equations of mathe- matical physics [8, 9, 35, 42]. The analytical methods used for solving these equa- tions are of minimal use. In general, such equations began to be studied at the end of the previous century and have been intensively developed in recent decades [11, 16, 19, 25, 30, 40, 44, 52]. One of the actively studied fractional diffusion equa- tions in recent years is the time-fractional diffusion equation which describes the mathematical processes of slow and super slow anomalous diffusion [15, 41, 51, 53]. But what about the diffusion equation with the time-fractional diffusion equation when the fractional operator includes a trace or combination of traces of the desired function? Here, we should also note the nonlocal problems [45] studied for the diffusion and wave equations, which have been successfully used in [5, 6, 10, 17, 26, 27, 39, 43] with applications to optimal control problems for dynamic populations. Such equations are also closely related to loaded equations [1, 12]. 2020 Mathematics Subject Classification. 35K15, 35R30. Key words and phrases. Heat equation; Cauchy problem; inverse problem; loaded equation; fractional operator. ©2024. This work is licensed under a CC BY 4.0 license. Submitted August 15, 2024. Published October 23, 2024. 1 2 R. P. AGARWAL, U. BALTAEVA, F. HUBERT, B. KHASANOV EJDE-2024/64 If we dwell, especially on problems for the loaded equation of parabolic type [45], or diffusion equation with the loaded term, then it should be noted the first results in these directions were obtained in the work of Nakhushev, Borisov, and Kreferov. Further, in [36] we find studies on the existence and uniqueness for the problem for a nonlinear loaded parabolic equation. Also in [24] we find studies of the Cauchy problem and related inverse problems for a one-dimensional nonlinear loaded parabolic equation in a special form. Also in [7, 29] there are studies of the initial and boundary value problems (Cauchy problem, Cauchy-Dirichlet problem) for equations of essential loaded par- abolic type loaded at a fixed time variable. A feature of these problems is the presence of a loaded term in an equation with a derivative of any integer order of the desired solution. In [37, 48] there are studies of boundary value problems for a fractionally loaded heat equation in two and three-dimensional domains. There when the order of the derivative in the loaded term is less than the order of the differential part, the load point moves. We should also note that in the recent papers [1, 28, 31, 50] there are interesting mixed-type loaded equations, which in- clude parabolic equations with fractional operators in two and three-dimensional domains. Therefore, loaded fractional-diffusion equations with Riemann-Liouville or Caputo operators and parabolic type equations with fractional loads in three and higher dimensional domains need to be studied. This is for the completeness of the theory of fractional diffusion and integro-differential equations, and for their numerous applications. In this work, along with the Cauchy problem, we study the inverse problem of determining the coefficient for the fractional time-loaded equation, which has attracted some interest in inverse problems research. Here, we also note the work [21] in two-dimensional problem of determining the diffusion coefficient for the frac- tional time equation. In [33] there are studies of inverse problems for the perturbed fractional diffusion time equation with a final redefinition. In [33] we find an ex- plicit formula for solving the anomalous diffusion equation in a multidimensional space. Inverse problems for time-fractional diffusion equations with nonclassical conditions were investigated in [14, 34, 49]. The aim of this work is to study the existence and uniqueness of the solution of direct and inverse problems for the heat equation in a multidimensional domain. As far as we know, very few researchers have studied problems for fractional diffusion equations in a multidimensional domain and fractional load equations. In this work, we generalize the study to the heat equation with a fractional load and equations of convolution type [23] in an n-dimensional domain. 2. Cauchy problem for integro-differential heat equations with fractional load In this section, we prove the existence and uniqueness of a solution to the Cauchy problem for the integro-differential equation of heat dissipation loaded with variable coefficients. Before proceeding to the formulation of the problem, we give some definitions and propositions for Holder spaces [38]. We introduce the following notation: Let Rn be the n dimensional Euclidean space, x = (x1, . . . , xn) ∈ Rn; Rn T is (n + 1) dimensional Euclidean space, consisting of points (x, t), where x ∈ Rn and t ∈ (0, T ], T > 0; Rn−1 T = {(x′, t) : x′ ∈ Rn−1, 0 < t < T}. EJDE-2024/64 INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS 3 Let f(x) be a position function in Rn. Definition 2.1. If for any two points x1, x2 ∈ Rn, there is a positive constant A such that |f(x1)− f(x2)| ≤ A|x1 − x2|l, then, the function f is said to satisfy the Holder condition with exponent l. The class of functions satisfying this condition is denoted as H l(Rn). Definition 2.2. If for any given pair of values (x(1), t1) and (x(2), t2) in Rn T , with x(1) = x (1) 1 , x (1) 2 , . . . , x (1) n and x(2) = x (2) 1 , x (2) 2 , . . . , x (2) n , if holds |f(x(1), t1)− f(x(2), t2)| ≤ n∑ i=1 Ai|x(1)i − x (2) i |l +An+1|t1 − t2|l/2, where Ai is a positive constant and l ∈ (0, 1), then the function f(x, t) is said to satisfy the Holder condition with exponent l, l/2 on Rn T . The class of such functions is denoted as H l,l/2(Rn T ). Cauchy problem. Find a solution u(x, t) in the domain (x, t) ∈ Rn T of the loaded heat equation ut − a(t)∆u = λD−α 0t u(x ′, t) + ∫ t 0 k(x′, τ)u(x, t− τ)dτ, (x, t) ∈ Rn T , (2.1) that satisfies the condition u(x, t) ∣∣ t=0 = φ(x), x ∈ Rn, (2.2) where D−α 0t is the Riemann-Liouville fractional integral operator of order α defined by D−α 0t u(x ′, t) = 1 Γ(α) ∫ t 0 (t− τ)α−1u(x′, τ)dτ, α > 0, a(t) ∈ E := {a(t) ∈ C1[0, T ] : 0 < a0 < a(t) ≤ a1 <∞}, λ ∈ R, ∆ := ∑n i=1 ∂2 ∂x2 i is the Laplace operator acting on variables x(x1, x2, . . . , xn), k(x ′, t), and φ(x) is a given real-valued function sufficiently smooth. Theorem 2.3. If a(t) ∈ E for all t ∈ (0, T ], and k(x′, t) ∈ H l,l/2(R n−1 T ), φ(x) ∈ H l+2(Rn), φ(x) ≤ φ0, where φ0 is a positive constant, then there exists a unique solution to the Cauchy problem in the domain u(x, t) ∈ H l+2,(l+2)/2(Rn T ), where (R n−1 T ) = {(x′, t) : x′ ∈ Rn−1, 0 ≤ t ≤ T}, l ∈ (0, 1). Thus, on the basis of the given functions k(x′, t) and φ(x), we will consider finding the function u(x, t) from the integro-differential equation (2.1) with the initial condition (2.2), i.e., the Cauchy problem. Before proceeding to study the Cauchy problem (2.1) and (2.2), we present the well-known solution to the problem for the classical inhomogeneous heat equation: v(x, t) = ∫ Rn ϕ(ξ)G(x− ξ, θ(t))dξ + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn F (ξ, θ−1(τ))G(x− ξ, θ(t)− τ)dξ, (2.3) 4 R. P. AGARWAL, U. BALTAEVA, F. HUBERT, B. KHASANOV EJDE-2024/64 which is the solution of the Cauchy problem for the heat equation with a time- varying heat conduction coefficient, vt − a(t)∆v = F (x, t), x ∈ Rn, t > 0, v(x, 0) = ϕ(x), x ∈ Rn. In (2.3), θ−1(t) is the inverse functions of θ(t) = ∫ t 0 a(τ)dτ , and G(x− ξ, θ(t)− τ) is a fundamental solution of the differential operator with a variable coefficient ∂/∂t− a(t)∆, ξ = (ξ1, ξ2, . . . , ξn); see [20]. Using (2.3), taking into account the properties of the fundamental solution, we have that the Cauchy problem (2.1) and (2.2) is equivalently reduced to the loaded integral equation of Volterra type with the shift [45], u(x, t) = ∫ Rn φ(ξ)G(x− ξ, θ(t))dξ + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)u(ξ, θ−1(τ)− α) ×G(x− ξ, θ(t)− τ)dξdα + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1 × u(ξ′, β)G(x− ξ, θ(t)− τ)dβdξ, (2.4) where ξ′ = (ξ1, ξ2, . . . , ξn−1), |x|2 = x21 + x22 + · · · + x2n. Thus, the last integral in relation (2.4) is understood as λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn−1 ∫ θ−1(τ) 0 (θ−1(τ)−β)α−1u(ξ′, β)G̃(x′−ξ, θ(t)−τ)dβdξ′, i.e. with a loaded member, where G̃(x′ − ξ, θ(t)− τ) = ∫∞ −∞G(x− ξ, θ(t)− τ)dξn. For the solvability of the loaded integral equation (2.4), we use the theory of Volterra integral equations. Lemma 2.4. Let φ(x) ∈ H l+2(Rn), φ(x) ≤ φ0 and k(x′, t) ∈ H l,l/2(R n−1 T ). Then there exists a unique solution u(x, t) to the integral equation (2.4). Proof. Using the method of successive approximations for (2.4) we define the se- quence {uj(x, t)}∞j=0 as follows: u0(x, t) = ∫ Rn φ(ξ)G(x− ξ, θ(t))dξ, u1(x, t) = λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1u0(ξ ′, β) ×G(x− ξ, θ(t)− τ)dβdξ + ∫ θ(t) 0 dτ a(θ−1(t)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)u0(ξ, θ −1(τ)− α) ×G(x− ξ, θ(t)− τ)dξdα, EJDE-2024/64 INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS 5 u2(x, t) = λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1u1(ξ ′, β) ×G(x− ξ, θ(t)− τ)dβdξ + ∫ θ(t) 0 dτ a(θ−1(t)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)u1(ξ, θ −1(τ)− α) ×G(x− ξ, θ(t)− τ)dξdα . . . (2.5) uj(x, t) = λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1uj−1(ξ ′, β) ×G(x− ξ, θ(t)− τ)dβdξ + ∫ θ(t) 0 dτ a(θ−1(t)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)uj−1(ξ, θ −1(τ)− α) G(x− ξ, θ(t)− τ)dξdα, for (x, t) ∈ Rn T , j = 1, 2, . . . . Using φ0 = |φ(x)|l in Rn T , and ∫ Rn G(x− ξ, θ(t)− τ)dξ = 1, (2.6) we estimate the modulus of functions uj(x, t) defined above, |u0(x, t)|l+2,(l+2)/2 T ≤ ∫ Rn |φ(ξ)|l+2,(l+2)/2 T G(x− ξ, θ(t))dξ ≤ φ0, and |u1(x, t)|l+2,(l+2)/2 T ≤ |λ| Γ(α) ∫ θ(t) 0 dτ |a(θ−1(t))| ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1|u0(ξ′, β)|l+2,(l+2)/2 T ×G(x− ξ, θ(t)− τ)dβdξ + ∫ θ(t) 0 dτ |a(θ−1(t))| ∫ Rn ∫ θ−1(τ) 0 |k(ξ′, α)|l+2,(l+2)/2 T × |u0(ξ, θ−1(τ)− α)|l+2,(l+2)/2 T Gdξdα ≤ φ0 |λ| Γ(α) Tα α a1 a0 t 1! + φ0 a1k0T a0 t 1! = φ0( |λ|Tα Γ(α+ 1) + k0T ) a1 a0 t1 1! , where k0 := |k(x′, t)|l,l/2T . 6 R. P. AGARWAL, U. BALTAEVA, F. HUBERT, B. KHASANOV EJDE-2024/64 We also estimate the modulus for u2(x, t), . . . : |u2(x, t)|l+2,(l+2)/2 T ≤ |λ| Γ(α) ∫ θ(t) 0 dτ |a(θ−1(t))| ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1|u1(ξ′, β)|l+2,(l+2)/2 T ×G(x− ξ, θ(t)− τ)dβdξ + ∫ θ(t) 0 dτ |a(θ−1(t))| ∫ Rn ∫ θ−1(τ) 0 |k(ξ′, α)|l+2,(l+2)/2 T | × u1(ξ, θ −1(τ)− α)|l+2,(l+2)/2 T Gdξdα ≤ φ0 ( |λ| Γ(α) Tα α a1 a0 )2 t2 2! + φ0( a1k0T a0 )2 t2 2! = φ0 ( ( |λ|Tα Γ(α+ 1) )2 + (k0T ) 2 ) ( a1 a0 )2 t2 2! , (2.7) |uj(x, t)|l+2,(l+2)/2 T ≤ φ0 (( |λ| Γ(α) Tα α )j + (k0T ) j )(a1 a0 )j tj j! , . . . As a result, we have the functional series ∞∑ j=0 uj(x, t). (2.8) Using the above estimates, according to the Weierstrass theorem on the smooth approximation of functional series [4], we can easily see that the obtained functional series converges. Using the definete integral for uj(x, t), the sequence of functions (2.8) converges uniformly to a function u(x, t) defined in H l+2,(l+2)/2(Rn T ). Thus, we have shown that there exists a solution of the integral equation (2.4), i.e., there is a solution to the Cauchy problem (2.1)-(2.2) as a mapping H l+2,(l+2)/2 → (Rn T ). Next we prove the uniqueness of this solution. Suppose on the contrary that the integral equation (2.4) has two different solutions u(1)(x, t) and u(2)(x, t): u(1)(x, t) = ∫ Rn φ(ξ)G(x− ξ, θ(t))dξ + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) × ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1u(1)(ξ′, β)G(x− ξ, θ(t)− τ)dβdξ + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)u(1)(ξ, θ−1(τ)− α) ×G(x− ξ, θ(t)− τ)dαdξ, and u(2)(x, t) = ∫ Rn φ(ξ)G(x− ξ, θ(t))dξ + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) × ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1u(2)(ξ′, β)G(x− ξ, θ(t)− τ)dβdξ EJDE-2024/64 INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS 7 + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)u(2)(ξ, θ−1(τ)− α) ×G(x− ξ, θ(t)− τ)dαdξ. Let the difference between these two functions be Z(x, t) = u(1)(x, t)− u(2)(x, t). Thus Z(x, t) satisfies a homogeneous integral equation, and Z(x, t) = λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1Z(ξ′, β) ×G(x− ξ, θ(t)− τ)dβdξ + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)Z(ξ, θ−1(τ)− α) ×G(x− ξ, θ(t)− τ)dξdα. (2.9) For t ∈ [0, T ] and x ∈ Rn the modular supremum of Z(x, t) is Z = sup |Z(x, t)|, t ∈ [0, T ]. In this case, it is easy to see that the following integral inequality holds Z(t) ≤ a1 a0 ( |λ| Γ(α) Tα α + k0T ) ∫ a1t 0 Z(t)dτ. Therefore, by the Gronwall-Bellman inequality [13], the last integral inequality has a unique solution, i.e. Z̄(t) ≡ 0, for all t ∈ [0, T ]. From this, we obtain that Z(x, t) ≡ 0, i.e. u(1)(x, t) = u(2)(x, t) in R n T . Thus, the integral equation (2.4) has a unique solution, and thus we can conclude that the equivalent problem (2.1) and (2.2) also has a unique solution. Lemma 2.4 is proved. □ 3. Inverse problem for the heat equation with fractional load Problem 3.1. Find functions u(x, t) and k(x′, t) in the domains (x, t) ∈ Rn T and (x′, t) ∈ Rn−1 T respectively, which satisfy the equation ut − a(t)∆u = λD−α 0,t u(x̃, t) + ∫ t 0 k(x′, τ)u(x, t− τ)dτ, (x, t) ∈ Rn T , (3.1) and the initial and boundary value conditions u(x, t) ∣∣ t=0 = φ(x), x ∈ Rn, (3.2) u ∣∣ xn=0 = f(x′, t), (x′, t) ∈ R̄n−1 T , (3.3) where a(t) ∈ E := {a(t) ∈ C1[0, T ] : 0 < a0 < a(t) ≤ a1 <∞}, ∆ is the Laplace operator acting on variables x(x1, x2, . . . , xn) ∈ Rn, x̃ ∈ Rn−2, where xi = xj = 0, at i ̸= j (1 ≤ i, j ≤ n), λ ∈ R, D−α 0t is the Riemann-Liouville fractional integral operator of order α (α > 0), φ(x) and f(x′, t) are given real- valued functions with f(x′, 0) = φ(x′, 0). 8 R. P. AGARWAL, U. BALTAEVA, F. HUBERT, B. KHASANOV EJDE-2024/64 We use the following notation (R n−2 T ) = {(x̃, t) : x̃ ∈ Rn−2, 0 ≤ t ≤ T}, (R n−1 T ) = {(x′, t) : x′ ∈ Rn−1, 0 ≤ t ≤ T}, l ∈ (0, 1). Remark 3.2. In loaded equations [28], a loaded operator or loaded term must include the trace of the desired function in the manifolds of dimension less than one from the sought-for solution in Rn+1 [3]. Based on this, we assume let x̃ ∈ Rn−2. (We choose so that there is a difference between the x′ functions). Thus, the problem can be continued in the same way in the cases where the loaded operatorMu ≡ λD−α 0,t u(x̃, t) contains the trace of the desired function, from the domains Rn−3, . . . , R1, i.e. respectively when x1 = 0, x2 = 0, . . . , xn = 0. Theorem 3.3. If f(x′, t) ∈ H l+4,(l+4)/2(R̄n−1 T ), f(x′, 0) = φ(x′, 0) and φ(x) ∈ H l+2(Rn), φ(x) ≤ φ0 = const > 0, then there exists a unique solution to the inverse problem in the domains Rn T and Rn−1 T respectively. This theorem will be proven using the theory of integral equations. First, we show that the inverse problem is equivalent to a system integral equations of Volterra type. We will introduce a new function as ϑ(x, t). Thus, by replacing ϑ(x, t) = uxnxn (x, t), then the problem (3.1)-(3.2) takes and follows (is taken and followed) the form: ϑt − a(t)∆ϑ = λD−α 0,t ϑ(x̃, t) + ∫ t 0 k(x′, τ)ϑ(x, t− τ)dτ, (3.4) ϑ(x, t) ∣∣ t=0 = φxnxn (x), (3.5) when xn = 0. Taking into account (3.1) and (3.2), we have an additional boundary condition (3.3) for the function ϑ(x, t) in the form ϑ(x, t) ∣∣ xn=0 = 1 a(t) ft(x ′, t)− n−1∑ k=1 ∂2 ∂x2k f(x′, t)− 1 a(t) ∫ t 0 k(x′, τ)f(x′, t− τ)dτ − λ a(t) D−α 0,t f(0, 0, x3 . . . , xn−1, t). (3.6) As a result, we obtain the following condition, in agreement with initial and bound- ary conditions (3.5) and (3.6): φxnxn(x ′, 0) = 1 a(0) ft(x ′, 0)− n−1∑ k=1 ∂2 ∂x2k f(x′, 0) (3.7) If matching conditions (3.6) and (3.7) are satisfied, f(x′, t) ∈ H l+4,(l+4)/2(R̄n−1 T ), f(x′, 0) = φ(x′, 0), and φ(x) ∈ H l+2(Rn), then it is equivalent to the inverse problem with respect to the function ϑ(x, t) = uxnxn(x, t), where u(x, t) = f(x′, t) + xnφn(x ′, 0) + ∫ xn 0 (xn − ξ)ϑ(x′, ξ, t)dξ. (3.8) EJDE-2024/64 INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS 9 Taking into account the matching conditions f(x′, 0) = φ(x′, 0) and (3.7), for t = 0, we obtain the following conditions u(x, t)|t=0 = f(x′, 0) + xnuxn (x′, 0, 0) + ∫ xn 0 (xn − ξ)φξξ(x ′, ξ)dξ = f(x′, 0) + xnuxn(x ′, 0, 0) + ∫ xn 0 (xn − ξ)dφξ = f(x′, 0) + xnuxn (x′, 0, 0) + (xn − ξ)φξ(x ′, ξ)|xn 0 − ∫ xn 0 φξ(x ′, ξ)dξ = f(x′, 0) + xnuxn (x′, 0, 0)− xnφxn (x′, 0) + φ(x)− φ(x′, 0) = xn(uxn(x ′, 0, 0)− φxn(x ′, 0)) + φ(x) = φ(x). It is easy to see from (3.8) that at xn = 0, our additional condition arises. Similarly, following the derivation of equation (3.1) from equation (3.4) as follows. Integrating twice from both parts of equation (3.4) from 0 to xn, we obtain∫ xn 0 (xn − ξ)ϑt(x ′, ξ, t)dξ − a(t) ∫ xn 0 (xn − ξ)∆ϑ(x′, ξ, t)dξ = ∫ xn 0 (xn − ξ) ∫ t 0 k(x′, τ)ϑ(x′, ξ, t− τ)dτdξ + λ Γ(α) ∫ xn 0 (xn − ξ) ∫ t 0 (t− τ)α−1u(x̃, τ)dτdξ. Therefore, taking into account the equality∫ xn 0 (xn − ξ)ϑ(x′, ξ, t)dξ = u(x, t)− f(x′, t)− xnφn(x ′, 0), we have ∂ ∂t (u(x, t)− f(x′, t)− xnφn(x ′, 0))− a(t)∆x′(u(x, t) − f(x′, t)− xnφn(x ′, 0))− a(t) ∫ xn 0 (xn − ξ)ϑxnxn (x′, ξ, t)dξ = ∫ t 0 k(x′, τ)(u(x, t− τ)− f(x′, t− τ)− xnφn(x ′, 0))dτ + λ Γ(α) ∫ t 0 (t− τ)α−1(u(x̃, τ)− f(x̃, t− τ)− xnφn(x ′, 0))dτ, i.e. ut(x, t)− ft(x ′, t)− a(t)∆x′u(x, t) + a(t)f(x′, t) + a(t)xnϑxn(x ′, 0, t) − a(t)ϑ(x, t) + a(t)ϑ(x′, 0, t) = ∫ t 0 k(x′, τ)u(x, t− τ)dτ − ∫ t 0 k(x′, τ)f(x′, t− τ)dτ − xnφn(x ′, 0) ∫ t 0 k(x′, τ)dτ + λ Γ(α) ∫ t 0 (t− τ)α−1u(x̃, τ)dτ − λ Γ(α) ∫ t 0 (t− τ)α−1(f(x̃, t− τ) + xnφn(x ′, 0))dτ. 10 R. P. AGARWAL, U. BALTAEVA, F. HUBERT, B. KHASANOV EJDE-2024/64 Thus, the inverse problem of (3.1)-(3.3) of finding the functions u(x, t) and k(x′, t) is equivalent to the problem of determining the functions ϑ(x, t) and k(x′, t) from problem (3.4)-(3.6). In the next step, equations (3.4), (3.6) with differentiated once by the variable t, and ϑt(x, t) = ω(x, t), as a result, we have (3.4), (3.5) and the auxiliary problems ωt − a(t)∆ω = (ln a(t))′ω − (ln a(t))′ ∫ t 0 k(x′, τ)ϑ(x, t− τ)dτ + ∫ t 0 k(x′, τ)ω(x, t− τ)dτ − λ(ln a(t))′D−α 0,t ϑ(x̃, t) + λD−α 0,t ω(x̃, t) + k(x′, t)φxnxn (x) + λ Γ(α) tα−1φxnxn (x̃), (3.9) and ω|t=0 = a(0)∆φxnxn (x), (3.10) ω|xn=0 = Ft(x ′, t) + a′(t) a2(t) ∫ t 0 k(x′, τ)f(x′, t− τ)dτ − 1 a(t) ∫ t 0 k(x′, τ)ft(x ′, t− τ)dτ − 1 a(t) k(x′, t)φ(x′, 0), (3.11) where F (x′, t) = 1 a(t) ft(x ′, t)− n−1∑ k=1 ∂2 ∂x2k f(x′, t) − λ a(t)Γ(α) ∫ t 0 (t− τ)α−1f(0, 0, x3, . . . , xn−1, τ)dτ. From relations (3.9), (3.10), and (3.11) we can find the functions ϑ(x, t), k(x′, t), and ω(x, t). When we integrate ϑt(x, t) = ω(x, t) from 0 to t, we obtain ϑ(x, t) = φxnxn(x) + ∫ t 0 ω(x, τ)dτ. (3.12) From this equality, if the function ω(x, t) is known, we can easily determine the function ϑ(x, t). Thus, from the problem (3.9)-(3.11) arises the problem (3.4)-(3.6), as a result of which (3.1)-(3.3) is passed to the inverse problem. Thus, the inverse problem of (3.1)-(3.3) of finding the functions u(x, t) and k(x′, t) is equivalent to the problem of determining the functions ϑ(x, t), ω(x, t) and k(x′, t) from problem (3.4)-(3.6) and (3.9) and (3.11). Thus, we proved the following lemma. Lemma 3.4. Let φ(x) ∈ H l+6(Rn), f(x′, t) ∈ H l+4,(l+4)/2(R̄n−1 T ), a(t) ∈ E and satisfies the agreement conditions f(x′, 0) = φ(x′, 0), φxnxn(x ′, 0) = 1 a(0) ft(x ′, 0)− n−1∑ k=1 ∂2 ∂x2k f(x′, 0), then (3.1), (3.2), (3.3) inverse problem is equivalent to the problem (3.4)-(3.6) and (3.9)-(3.11), finding functions ϑ(x, t), k(x′, t), and ω(x, t) In the next stage, we will transform the lemma on the equivalence of the obtained problem (3.4)-(3.6) and (3.9)-(3.11) into a system of loaded integral equations. EJDE-2024/64 INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS 11 Lemma 3.5. Equations (3.4)-(3.6) and (3.9)-(3.11) auxiliary problems, equivalent to finding functions ϑ(x, t), k(x′, t), ω(x, t), from the following system of integral equations: ϑ(x, t) = ψ01(x, t) + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn dξ × ∫ θ−1(τ) 0 k(ξ′, α)ϑ(ξ, θ−1(τ)− α)G(x− ξ, θ(t)− τ)dα + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ϑ(ξ̃, β) ×G(x− ξ, θ(t)− τ)dβdξ, (3.13) ω(x, t) = ψ02(x, t) + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn [ (ln a(θ−1(τ)))′ω(ξ, θ−1(τ)) − (ln a(θ−1(τ)))′ ∫ θ−1(τ) 0 k(ξ′, α)ϑ(ξ, θ−1(τ)− α)dα ] ×G(x− ξ, θ(t)− τ)dξ + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)ω(ξ, θ−1(τ)− α)G(x− ξ, θ(t)− τ)dαdξ + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn k(ξ′, θ−1(τ))φξnξn(ξ)G(x− ξ, θ(t)− τ)dξ + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ω(ξ̃, β)G(x− ξ, θ(t)− τ)dβdξ − λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn (ln a(θ−1(τ)))′ ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ϑ(ξ̃, β)Gdβdξ, (3.14) k(x′, t) = ψ03(x, t) + a(t) φ(x′, 0) {∫ θ(t) 0 dτ a(θ−1(τ)) × ∫ Rn ((ln a(θ−1(τ)))′ ∫ θ−1(τ) 0 k(ξ′, α)ϑ(ξ, θ−1(τ)− α)dα)G(x′ − ξ′, ξn, θ(t)− τ)dξ − ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn (ln a(θ−1(τ)))′ω(ξ, θ−1(τ))G(x′ − ξ′, ξn, θ(t)− τ)dξ − ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 k(ξ′, α)ω(ξ, θ−1(τ)− α)G(x′ − ξ′, ξn, θ(t)− τ)dαdξ − ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn k(ξ′, θ−1(τ))φξnξn(ξ)G(x ′ − ξ′, ξn, θ(t)− τ)dξ + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) × ∫ Rn ((ln a(θ−1(τ)))′ ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ϑ(ξ̃, β)dβG(x′ − ξ′, ξn, θ(t)− τ)dξ 12 R. P. AGARWAL, U. BALTAEVA, F. HUBERT, B. KHASANOV EJDE-2024/64 − ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ω(ξ̃, β)G(x′ − ξ′, ξn, θ(t)− τ)dβdξ) } + (ln a(t))′ φ(x′, 0) ∫ t 0 k(x′, τ)f(x′, t− τ)dτ − 1 φ(x′, 0) ∫ t 0 k(x′, τ)ft(x ′, t− τ)dτ, (3.15) where ψ01(x, t) = ∫ Rn φξnξn(ξ)G(x− ξ, θ(t))dξ, ψ02(x, t) = ∫ Rn a(0)∆φξnξn(ξ)G(x− ξ, θ(t))dξ + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn (θ−1(τ))α−1φξnξn(ξ̃)G(x− ξ, θ(t)− τ)dξ, ψ03(x, t) = a(t) φ(x′, 0) (Ft(x ′, t)− ∫ Rn a(0)∆φξnξn(ξ)G(x ′ − ξ′, ξn, θ(t))dξ) + a(t) φ(x′, 0) (− λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn (θ−1(τ))α−1φξnξn(ξ̃) ×G(x′ − ξ′, ξn, θ(t)− τ)dξ). The proof of the lemma 3.5 can obviously be obtained, from (3.4), (3.5), (3.9), and (3.10), we did the integral relations (3.13) and (3.14), as the integral equation (2.4). And also, from equations (3.11) and (3.14), we easily obtain relation (3.15). 4. Existence and uniqueness for the inverse problem In this section, we demonstrate one of the main results of the inverse problem, the existence and uniqueness theorem for solutions to the system of integral equations (3.13)-(3.15), from which the unique solvability of the problem (3.1)-(3.3) follows. Theorem 4.1. Let a(t) ∈ E, φ(x) ∈ H l+6(Rn), and f(x′, t) ∈ H l+4,(l+4)/2(R̄n−1 T ), moreover f(x′, 0) = φ(x′, 0)φxnxn (x′, 0) = 1 a(0) ft(x ′, 0)− n−1∑ k=1 ∂2 ∂x2k f(x′, 0), and the terms of the agreement be reasonable. Then there exists a sufficiently small number T0 > 0 such that for any T ∈ (0, T0], there is a unique solution to the system of integral equations (3.13)–(3.15), in the domains {ϑ(x, t), ω(x, t)} ∈ H l+2,(l+2)/2(R̄n T ), k(x ′, t) ∈ H l,l/2(R̄n−1 T ). Proof. Without loss of generality, we can prove this theorem by the classical integral method. First, we write the system of integral equations (3.13)-(3.15) in the form of an operator equation ψ = Lψ, (4.1) where ψ = (ψ1, ψ2, ψ3) ∗ = (ϑ(x, t), ω(x, t), k(x′, t))∗, with ∗ denote the transposi- tion. Equations (3.13)-(3.15) as operator in equations become Lψ = [(Lψ)1, (Lψ)2, (Lψ)3] ∗. EJDE-2024/64 INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS 13 Operators (Lψ)i, i = 1, 2, 3 can be written in the form (Lψ)1 = ψ01(x, t) + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 ψ3(ξ ′, α)ψ1(ξ, θ −1(τ)− α) ×G(x− ξ, θ(t)− τ)dαdξ + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) × ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ψ1(ξ̃, β)Gdβdξ, (4.2) (Lψ)2 = ψ02(x, t) + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn [ (ln a(θ−1(τ)))′ψ2(ξ, θ −1(τ)) − (ln a(θ−1(τ)))′ ∫ θ−1(τ) 0 ψ3(ξ ′, α)ψ1(ξ, θ −1(τ)− α)dα ] ×G(x− ξ, θ(t)− τ)dξ + ∫ θ(t) 0 dτ a(θ−1(τ)) × ∫ Rn ∫ θ−1(τ) 0 ψ3(ξ ′, α)ψ2(ξ, θ −1(τ)− α)G(x− ξ, θ(t)− τ)dαdξ + ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ψ3(ξ ′, θ−1(τ))φξnξn(ξ)G(x− ξ, θ(t)− τ)dξ + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ψ2(ξ̃, β)Gdβdξ − λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn (ln a(θ−1(τ)))′× × ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ψ1(ξ̃, β)G(x− ξ, θ(t)− τ)dβdξ, (4.3) (Lψ)3 = ψ03(x, t) + a(t) φ(x′, 0) (∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ln a(θ−1(τ)) )′ × ∫ θ−1(τ) 0 ψ3(ξ ′, α)ψ1(ξ, θ −1(τ)− α)Gdαdξ − ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ( ln a(θ−1(τ)) )′ ψ2(ξ, θ −1(τ))Gdξ − ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 ψ3(ξ ′, α)ψ2(ξ, θ −1(τ)− α)Gdαdξ − ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ψ3(ξ ′, θ−1(τ))φξnξn(ξ)G(x ′ − ξ′, ξn, θ(t)− τ)dξ − λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ψ2(ξ̃, β)Gdβdξ + λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ( ln a(θ−1(τ)) )′ 14 R. P. AGARWAL, U. BALTAEVA, F. HUBERT, B. KHASANOV EJDE-2024/64 × ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ψ1(ξ̃, β)Gdβdξ) + ( ln a(t) )′ φ(x′, 0) ∫ t 0 ψ3(x ′, τ)f(x′, t− τ)dτ − 1 φ(x′, 0) ∫ t 0 ψ3(x ′, τ)ft(x ′, t− τ)dτ. (4.4) Therefore, in relations (4.2)–(4.4), taking into account the notation of Lemma 3.5, we denote |ψ|l,l/2T = max(|ψ1|l.l/2T0 , |ψ2|l.l/2T0 , |ψ3|l.l/2T0 ), and on H l,l/2(Rn T ), we include the conditions S(T ) = |ψ1 − ψ0|l,l/2T ≤ |ψ0|l,l/2T0 , (4.5) where ψ0 = (ψ01, ψ02, ψ03) and |ψ0|l,l/2T0 = max(|ψ01|l,l/2T0 , |ψ02|l,l/2T0 , |ψ03|l,l/2T0 ). □ Suppose that ψ is an arbitrary in S(T ), here T < T0. In this case, the following inequalities are valid: |ψi|l,l/2T ≤ 2|ψ0|l,l/2T0 , i = 1, 2, 3. Therefore, it is similar to setting the Cauchy problem for the classical heat equation when we consider a function in the class φ(x) ∈ H l+6, and introduce the following notation: a2 := max t∈[0,T ] |(ln a(t))′|, φ1 := |φ|l+6, f0 := |f |l+4,(l+4)/2 T . Let the operator L be defined in a closed S, which is part of the Banach space. Definition 4.2. An operator L is called a contraction operator in S, when the following two conditions are satisfied: (1) if y ∈ S, then Ly ∈ S, i.e. the operator L maps the set S in itself; (2) if there exists a real number ρ ∈ [0, 1) such that ∥Ly − Lz∥ ≤ ρ ∥y − z∥ , ∀y, z ∈ S . If for the mapping L : X → X there exists a point x ∈ X such that Lx = x, then the point x is called the fixed point of L, [4]. In this section, we state and prove the contraction mapping principle [18], which is one of the most useful methods for the construction of the solution of the differ- ential equations [20]. Contraction mapping principle. Every contraction mapping defined in a com- plete metric space R has one, and only one fixed point, that is, the equation, Lx = x has a unique solution x0 ∈ S. First of all, we show that the operator L satisfies the first condition for the definition of compressibility in S, i.e.: |(Lψ)1 − ψ01|l,l/2T = ∣∣ ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 ψ3(ξ ′, α)ψ1(ξ, θ −1(τ)− α) EJDE-2024/64 INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS 15 ×G(x− ξ, θ(t)− τ)dαdξ ∣∣l,l/2 T + ∣∣ λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) × ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ψ1(ξ̃, β)G(x− ξ, θ(t)− τ)dβdξ ∣∣l,l/2 T , here after replacing |y = θ−1(τ)| as a result, we obtain∣∣ ∫ t 0 dτ a(θ−1(τ)) ∫ Rn ∫ y 0 ψ3(ξ ′, α)ψ1(ξ, y − α)G(x− ξ, θ(t)− θ(y))dαdξ ∣∣l,l/2 T + ∣∣ λ Γ(α) ∫ t 0 dτ a(θ−1(τ)) ∫ Rn ∫ y 0 (y − β)α−1ψ1(ξ̃, β) ×G(x− ξ, θ(t)− θ(y))dβdξ|l,l/2T ≤ β̄0(T ) ∣∣(ψ3(ξ ′, t0)ψ1(ξ, y − t0)) ∣∣l,l/2 T + β̄1(T )|(ψ1(ξ̃, t0)) ∣∣l,l/2 T ≤ 4β0(T )(|ψ0|l,l/2T0 )2 + 2β1(T )(|ψ0|l,l/2T0 ), Similarly, we obtain estimates for other components of the vector L:∣∣(Lψ)2 − ψ02 ∣∣l,l/2 T ≤ 4β1(T )(a2 + 1)(|ψ0|l,l/2T0 )2 + 2β2(T )(2a2 + φ1 + 1)(|ψ0|l,l/2T0 ),∣∣(Lψ)3 − ψ03 ∣∣l,l/2 T ≤ 2(β1(T )a1φ −1 0 (2a2 + φ1 + 1) + f0φ −1 0 T0(a2 + 1))|ψ0|l,l/2T0 + 4β2(T )a1φ −1 0 (a2 + 1)(|ψ0|l,l/2T0 )2. Here, If T → 0 then, βi(T ), (i = 0, 1, 2) tends to zero. If we choose T0 such that 4β0(T0)(|ψ0|l,l/2T0 )2 + 2β1(T0)(|ψ0|l,l/2T0 ) ≤ 1, 4β1(T0)(a2 + 1)(|ψ0|l,l/2T0 )2 + 2β2(T0)(2a2 + φ1 + 1)(|ψ0|l,l/2T0 ) ≤ 1, 2(β1(T0)a1φ −1 0 (2a2 + φ1 + 1) + f0φ −1 0 T0(a2 + 1))|ψ0|l,l/2T0 + 4β2(T0)a1φ −1 0 (a2 + 1)(|ψ0|l,l/2T0 )2 ≤ 1, (4.6) then the operator L satisfies the first mapping condition by reducing T < T0, i.e., LS ⊂ S. In the same way, we can pass the second condition to the contraction principle. We suppose that ψ(1) = (ψ (1) 1 , ψ (1) 2 , ψ (1) 3 ) ∈ S(T ), ψ(2) = (ψ (2) 1 , ψ (2) 2 , ψ (2) 3 ) ∈ S(T ). To estimate the distance between the images of the functions ψ(1) and ψ(2) as a result of mapping L, with regards to∣∣ψ(1) 2 ψ (1) 1 − ψ (2) 2 ψ (2) 1 ∣∣l,l/2 T = |(ψ(1) 2 − ψ (2) 2 )ψ (1) 1 + ψ (2) 2 (ψ (1) 1 − ψ (2) 1 )|l,l/2T ≤ 2|ψ(1) − ψ(2)|l,l/2T max(|ψ(1) 1 |l,l/2T , |ψ(2) 2 |l,l/2T ) ≤ 4|ψ0|l,l/2T |ψ(1) − ψ(2)|l,l/2T , we have |((Lψ)(1) − (Lψ)(2))1|l,l/2T = ∣∣ ∫ θ(t) 0 dτ a(θ−1(τ)) ∫ Rn ∫ θ−1(τ) 0 ψ3(ξ ′, α)ψ1(ξ, θ −1(τ)− α) ×G(x− ξ, θ(t)− τ)dαdξ|l,l/2T + | λ Γ(α) ∫ θ(t) 0 dτ a(θ−1(τ)) 16 R. P. AGARWAL, U. BALTAEVA, F. HUBERT, B. KHASANOV EJDE-2024/64 × ∫ Rn ∫ θ−1(τ) 0 (θ−1(τ)− β)α−1ψ1(ξ̃, β)G(x− ξ, θ(t)− τ)dβdξ ∣∣l,l/2 T ≤ [8β0(T )|ψ0|l,l/2T0 + 4β1(T )]|ψ(1) − ψ(2)|l,l/2T . In the same way, the second and third components of Lψ satisfy:∣∣((Lψ)(1) − (Lψ)(2))2 ∣∣l,l/2 T ≤ [2β1(T )(2a2 + φ1 + 1) + 8β2(T )(a2 + 1)|ψ0|l,l/2T0 ]|ψ(1) − ψ(2)|l,l/2T0 , |((Lψ)(1) − (Lψ)(2))3|l,l/2T ≤ [2(β1(T )a1φ −1 0 (2a2 + φ1 + 1) + f0φ −1 0 T0(a2 + 1))]|ψ(1) − ψ(2)|l,l/2T0 + [8β2(T )a1φ −1 0 (a2 + 1)(|ψ0|l,l/2T0 )]|ψ(1) − ψ(2)|l,l/2T0 . Therefore, |(Lψ(1) − Lψ(2))|l,l/2T < ρ|ψ(1) − ψ(2)|l,l/2T . Thus, if the following conditions are satisfied [8β0(T )|ψ0|l,l/2T0 + 2β1(T )] ≤ ρ < 1, [2β1(T )(2a2 + φ1 + 1) + 8β2(T )(a2 + 1)|ψ0|l,l/2T0 ] ≤ ρ < 1, [2(β1(T )a1φ −1 0 (2a2 + φ1 + 1) + f0φ −1 0 T0(a2 + 1))] + [8β2(T )a1φ −1 0 (a2 + 1)(|ψ0|l,l/2T0 )] ≤ ρ < 1, (4.7) then, operator L is compact on S(T ), [20]. From the fulfillment of inequality (4.7), it is not difficult to see that condition (4.6) is satisfied for T0. It follows that the contraction principle holds for in T < T0. In this case, according to the Banach fixed point theorem [4], there exists a unique solution to the equation (4.1). Therefore, from the system of integral equations (3.13)-(3.15), using the method of successive approximations, we determine the unique solution of the system that belongs to the class H l+2,(l+2)/2(R̄n T ). 5. Conclusion Parabolic equations, especially diffusion and reaction-diffusion equations, are one of the most used equations of the modern theory of partial differential equations, which have been rapidly developing in recent years. This happens because the mathematical models for the dynamics of infectious diseases can show optimal con- trol of the spread, and controllability and stabilization of the dynamics of infection. Based on this, in this paper we study the solvability of direct and inverse problems for a multidimensional fractionally loaded heat equation in Holder spaces. Introductory section of this work is dedicated to the history and literature on problems for the fractional differential equations, time-fractional diffusion equa- tions, integro-differential equations with fractional load, and the novelty of the work. In the second section, we investigated the Cauchy problem for the integro- differential equation with a fractional load. We proved the unique solvability of the initial-value problem using the theory of the integral equations and the meth- ods of successive approximations. EJDE-2024/64 INVERSE PROBLEMS FOR INTEGRO-DIFFERENTIAL HEAT EQUATIONS 17 In the third section, we formulate inverse problems. With a change of variables, the formulated problem (3.1)-(3.3) is reduced to auxiliary problems with new func- tions v(x, t), w(x, t) and k(x′, t). Hence, the auxiliary problems are equivalently reduced to a system of loaded integral equations of Volterra type. 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Lett., 91 (2019), 15-21. [52] Yang, Xiao-Jun; Baleanu,Dumitru; Srivastava, H. M.; Local fractional integral transforms and their applications. Elsevier 2015. [53] Zhang, Wei; Cai, Xing; Holm, Sverre; Time-fractional heat equations and negative absolute temperatures, Computers and Mathematics with Applications, Volume 6 (2014) 7, Issue 1, 164-171 (2014). Ravi P. Agarwal Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA Email address: agarwalr@fit.edu Umida Baltaeva Khorezm Mamun Academy, Khorezm, Uzbekistan. Urgench State University, Uzbekistan Email address: umida baltayeva@mail.ru Florence Hubert Aix-Marseille Universite, CNRS, I2M, Marseille, France Email address: florence.hubert@univ-amu.fr Boburjon Khasanov Khorezm Mamun Academy, Khorezm, Uzbekistan Email address: xasanovboburjon.1993@gmail.com 1. Introduction 2. Cauchy problem for integro-differential heat equations with fractional load Cauchy problem. 3. Inverse problem for the heat equation with fractional load 4. Existence and uniqueness for the inverse problem Contraction mapping principle 5. Conclusion References