# American Institute of Mathematical Sciences

November  2014, 13(6): 2509-2542. doi: 10.3934/cpaa.2014.13.2509

## A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach

 1 Dipartimento di Matematica, Universitá degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy, Italy

Received  January 2014 Revised  June 2014 Published  July 2014

We consider a heat transmission problem for a composite material which fills the $n$-dimensional Euclidean space. The composite has a periodic structure and consists of two materials. In each periodicity cell one material occupies a cavity of size $\epsilon$, and the second material fills the remaining part of the cell. We assume that the thermal conductivities of the materials depend nonlinearly upon the temperature. We show that for $\epsilon$ small enough the problem has a solution, \textit{i.e.}, a pair of functions which determine the temperature distribution in the two materials. Then we analyze the behavior of such a solution as $\epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis. In particular we prove that if $n\geq 3$, the temperature can be expanded into a convergent series expansion of powers of $\epsilon$ and that if $n=2$ the temperature can be expanded into a convergent double series expansion of powers of $\epsilon$ and $\epsilon \log \epsilon$.
Citation: Massimo Lanza de Cristoforis, aolo Musolino. A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2509-2542. doi: 10.3934/cpaa.2014.13.2509
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