Global solution to a non-classical heat problem in the semi-space ℝ⁺×ℝⁿ⁻¹
Mahdi Boukrouche, Domingo Alberto Tarzia · Quarterly of Applied Mathematics · 2014
We consider the non-classical heat equation in the n n -dimensional domain D = R + × R n − 1 D=\mathbb {R}^{+}\times \mathbb {R}^{n-1} for which the internal energy supply depends on the heat flux on the boundary S = ∂ D S=\partial D . The problem is motivated by the modeling of temperature regulation in the medium. Using the Green function for the domain D D , the solution is found for an integral representation depending on the heat flux V V on S S which is an additional unknown of the problem. We obtain that V V must satisfy a Volterra integral equation of second kind at time t t with a parameter in R n − 1 \mathbb {R}^{n-1} . Under some conditions on data, we show that there exists a unique local solution which can be extended globally in time. This work generalizes the results obtained in the one-dimensional case.