A heat conduction problem with sources depending on the average of the heat flux on the boundary

M. BOUKROUCHE; D. A. TARZIA

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Lugar: Bahia Blanca; Año: 2020 vol. 61 p. 87 - 101

0041-6932

Motivated by the modeling of temperature regulation in some mediums,we consider the non-classical heat conduction equation in the domain $D=r^{n-1}imesr^{+}$ for which theinternal energy supply depends on an average in the time variable%${1over t}int_{0}^{t} u_{x}(0 , y , s) ds$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 $r^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which canbe 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.