A Heat Conduction Problem with Sources Depending on the Average of the Heat Flux on the Boundary

Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0 , y , s)$ on the boundary $S=\partial D$. The solution to the problem is found for an integral representation depending on the heat flux on $S$ which is an additional unknown of the considered problem. We obtain that the heat flux $V$ must satisfy a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method.


Introduction
Let 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 problem 1.1 on the non-classical heat equation, in the semi-n-dimensional space domain D with non local sources, for which the internal energy supply depends on the average 1 t t 0 u x (0, y, s)ds of the heat flux on the boundary S. Problem 1.1. Find the temperature u at (x, y, t) satisfying the following conditions x > 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), where ∆ denotes the Laplacian in R n . This problem is motivated by modeling the temperature in an isotropic medium with the average of 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 (y, s) → V (y, s) = u x (0, y, s) at the boundary S, see [11,13]. Some references on the subject are [6] where F 1 t t 0 u x (0, y, s)ds is replaced by F (u x (0, y, t)), or [7] where is replaced by F t 0 u x (0, y, s)ds ; see also [4], [14], [23], [24] where the semi-infinite case of this nonlinear problem with F (u x (0, y, t)) have been considered. The non-classical one-dimensional heat equation in a slab with fixed or moving boundaries was studied in [14], [22]. See also other references on the subject [8]- [10], [12], [16]- [19]. To our knowledge, it is the first time that the solution to the average of a non-classical heat conduction of the type of Problem 1.1 is given. Other non-classical problems can be found in [5].
In [6] basic solution to the n-dimensional heat equation, and a technical Lemma was established. We prove in Section 2 the local existence of a solution for the considered Problem 1.1 under some conditions on data F and h which can be extended globally in times. Moreover, in Section 3 we consider the corresponding one dimensional problem and we obtain its explicit solution for the heat flux and the average of the total heat flux at the face x = 0, by using the Laplace transform and also the Adomian decomposition method [1,2,3,7,25,26].

Existence results
In this Section, we give first in Theorem 2.1, the integral representation (2.2) of the solution of the considered Problem 1.1, but it depends on the heat flow V on the boundary S, which satisfies the Volterra integral equation (2.4) with initial condition (2.5). Then we prove, in Theorem 2.3, under some assumptions on the data, that there exists a unique solution of the problem locally in times which can be extended globally in times.
We first recall here the Green's function for the n-dimensional heat equation with homogenuous Dirichlet's boundary conditions, given the following expression where G is the Green's function for the one-dimensional case given by Theorem 2.1. The integral representation of a solution of the Problem 1.1 is given by the following expression and the heat flux (y, t) → V (y, t) = u x (0, y, t) on the surface x = 0, satisfies the following Volterra integral equation Proof. As the boundary condition in Problem (1.1) is homogeneous, we have from [15,20] u(x, y, t) = 4Problem with Sources Depending on the Average of the Heat Flux on the Boundary and therefore Also by (2.1) we obtain Taking this formula in (2.6) we obtain (2.2).
Proof. Using the derivative, with respect to x, of (2.1), then taking x = 0 and τ = 0, then taking the new expression of V 0 (y, t) in the Volterra integral equation (2.4) we obtain (2.9). Proof. We know from Theorem (2.1) that, to prove the existence and uniqueness of the solution (2.2) of Problem (1.1), it is enough to solve the Volterra integral equation (2.9). So we rewrite it as follows and So 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 [21].
• The function f is defined and continuous for all (y, t) ∈ R n−1 × R + , so H1 holds. 6Problem with Sources Depending on the Average of the Heat Flux on the Boundary • 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.
is C 1 (R + ) and is in the compact B ⊂ R for all z ∈ R n−1 , so by the continuity of F we get F 1 we obtain .
• For all compact I ⊂ R + , for all function ψ ∈ C(I, R n ), and all t 0 > 0, with Sources Depending on the Average of the Heat Flux on the Boundary as F ∈ C(R) and ψ ∈ C(I, R n ) then there exists a constant M > 0 such that Then we obtain as for H4, that lim t→t0 I |g(t, τ ; ψ(τ )) − g(t 0 , τ, ψ(τ ))|dτ = 0.
• 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 . We have also for each t ∈ [0, k] the function ϕ ∈ L 1 (0, t) as a function of τ and we have also 3 The one-dimensional case of Problem 1.1 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 V (t) = u x (0, t) is the the solution of the following Volterra integral equation of the second kind For the particular case h(x) = h 0 > 0 pour x > 0, and F (V ) = λV pour λ ∈ R (3.6) then we have and the integral equation (3.4) becomes where W (t) is defined by By using the integral equation (3.8) for V (t) we obtain for W (t) the following Volterra integral equation of the second kind: (3.11) by using that Therefore, we deduce the following results Theorem 3.1. Taking h and F as in (3.6), the solution of the non-classical heat conduction Problem 3.1 is given by (3.9) where W (t) is the solution of the Volterra integral equation (3.11). Moreover, its Laplace transform L is given by the following expression:

Conclusion:
We have obtained the global solution of a non-classical heat conduction problem in a semi-n-dimensional space, in which the source depends of the average of the total heat flux on the face x = 0. Moreover, for the one-dimensional case we have obtained the explicit solution by using the Laplace transform and also the Adomian decomposition method.