Non-classical heat conduction problem with non local source

We consider the non-classical heat conduction equation, in the domain $D=\br^{n-1}\times\br^{+}$, for which the internal energy supply depends on an integral function in the time variable of % $(y , t)\mapsto \int_{0}^{t} u_{x}(0 , y , s) ds$, %where $u_{x}(0 , y , s)$ is the heat flux on the boundary $S=\partial D$, with homogeneous Dirichlet boundary condition and an initial condition. The problem is motivated by the modeling of temperature regulation in the medium. The solution to the problem is found using a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\br^{n-1}$. The solution to this Volterra equation is the heat flux $(y, s)\mapsto V(y , t)= u_{x}(0 , y , t)$ on $S$, which is an additional unknown of the considered problem. We show that a unique local solution exists, which can be extended globally in time. Finally a one-dimensional case is studied with some simplifications, we obtain the solution explicitly by using the Adomian method and we derive its properties.


Introduction
Let's consider the domain D and its boundary S defined by D = R n−1 × R + = {(x, y) ∈ R n : x = x 1 > 0, y = (x 2 , · · · , x n ) ∈ R n−1 }, (1.1) S = ∂D = R n−1 × {0} = {(x, y) ∈ R n : x = 0, y ∈ R n−1 }. (1. 2) The aim of this paper is to study the following the problem 1.1 with the non-classical heat equation, in the domain D with non local source, for which the internal energy supply depends on the integral t 0 u x (0, y, s)ds on the boundary S. Problem 1.1. Find the temperature u, at (x, y, t) such that it satisfies the following conditions u t − ∆u = −F t 0 u x (0, y, s)ds , x = x 1 > 0, y ∈ R n−1 , t > 0, u(0, y, t) = 0, y ∈ R n−1 , t > 0, u(x, y, 0) = h(x, y), x > 0, y ∈ R n−1 , where ∆ denotes the Laplacian in R n . This problem is motivated by the modeling of temperature regulation in an isotropic medium, with non-uniform and non local sources that provide cooling or heating system. According to the properties of the function F with respect to the heat flow V (y, s) = u x (0, y, s) at the boundary S. For example, assuming that V F(V ) > 0, ∀V = 0, F(0) = 0, (1.3) with F(V (y, t)) = F t 0 V (y, s)ds (1.4) then, see [12,14], the cooling source occurs when V (y, t) > 0 and heating source occurs when V (y, t) < 0. Some references on the subject are [8] where F(V ) = F (V ), [5,15,27,28] where the following semi-one-dimension of this nonlinear problem, have been considered. The nonclassical one-dimensional heat equation in a slab with fixed or moving boundaries was studied in [9,10,11,25]. More references on the subject can be found in [13,18,19,21,22]. To our knowledge, it is the first time that the solution to a non-classical heat conduction of the type of Problem 1.1 is given. Other non-classical problems can be found in [6].
The goal of this paper is to obtain in Section 2 the existence and the uniqueness of the global solution of the non-classical heat conduction Problem 1.1, which is given through a Volterra integral equation. In Section 3 we obtain the explicit solution of the one-dimensional case of Problem 1.1, with some simplifications, which is obtained by using the Adomian method through a double induction principle.
We recall here the Green's function for the n-dimensional heat equation with homogeneous Dirichlet's boundary conditions, given the following expression [17,23] where G is the Green's function for the one-dimensional case given by

Existence results
In this Section, we give first in Theorem 2.1, the integral representation (2.
is the error function, with and the heat flux V (y, t) = u x (0, y, t) on the surface x = 0, satisfies the following Volterra integral equation in the variable t > 0, with y ∈ R n−1 is a parameter and where the function (y, t) → F(V (y, t)) is defined by (1.4) for y ∈ R n−1 and t > 0.
Also by (1.5) we obtain and by using Taking this formula in (2.5) we obtain (2.1).
Proof. Using the derivative, with respect to x, of (1.5), then taking x = 0 and τ = 0, then taking the new expression of V 0 (y, t) in the Volterra integral equation ( and We have to check the conditions H1 to H4 in Theorem 1.1 page 87, and H5 and H6 in Theorem 1.2 page 91 in [24]. • The function f is defined and continuous for all (y, t) ∈ R n−1 × R + , so H1 holds.
• The function g is measurable in (t, τ, y, x) for 0 ≤ τ ≤ t < +∞, x ∈ R, y ∈ R n−1 , and continuous in x for all (y, t, τ ) ∈ R n−1 × R + × R + , g(y, t, τ, x) = 0 if τ > t, so here we need the continuity of which follows from the hypothesis that F ∈ C(R). So H2 holds.
• For all k > 0 and all bounded set B in R, we have thus there exists a measurable function m given by and satisfies so H3 holds.
is C 1 (R) and is in the compact B ⊂ R for all η ∈ R n−1 , so by the continuity of F we get F(V (η, τ )) ⊂ F(B), that is there exists M > 0 such that F(V (η, τ ))| ≤ M for all (η, τ ) ∈ R n−1 × R + . So .
• Now for each constant K > 0 and each bounded set B ⊂ R n−1 there exists a measurable function ϕ such that |g(y, t, τ, x) − g(y, t, τ, X)| ≤ ϕ(t, τ )|x − X| whenever 0 ≤ τ ≤ t ≤ K and both x and X are in B. Indeed as F is assumed locally Lipschitz function in R there exists constant L > 0 such that with L(τ ) = Lτ . Then we have |g(y, t, τ, x) − g(y, t, τ, X)| = 2 . We have also for each t ∈ [0, k] the function ϕ ∈ L 1 (0, t) as a function of τ and  Let us consider now the one dimensional case of Problem 1.1 for the temperature defined by Problem 3.1. Find the temperature u at (x, t) such that it satisfies the following conditions Taking into account that thus the solution of the problem 3.1 is given by and W (t) = u x (0, t) is the the solution of the following Volerra integral equation For the particular case then we have and the integral equation (3.4) becomes Lemma 3.1. Assume (3.6) holds. The solution of problem 3.1 is given by where U is given by and g is the solution of the Volterra integral equation Moreover, the heat flux on x = 0 is given by Proof. We set thus the function U satisfies the following new Volterra integral equation by using the following equality From [ [4], p.229], the solution t → U (t) of the integral equation (3.14) is given by (3.10) where g is the solution of the Volterra equation (3.11). From (3.11) we obtain that using the following equality where B and Γ are the classical Beta and Gamma functions defined below. Therefore, we have that (3.18) and then the heat flux on x = 0 is given by u x (0, t) = W (t) = U ′ (t), that is (3.12) holds.
We recall here the well known Beta an Gamma functions defined respectively by We will use in the next theorem the well known relations and in particular the following one Lemma 3.2. For all integer n ≥ 1 we have and we use the definition for compactness expression.
We need previously some preliminary simple results in order to obtain the solution of the integral equation   (3(2n + 1))!! (3.27) are series with infinite radii of convergence.