A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. (English) Zbl 1304.35686
Summary: 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, 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 \(\epsilon \log \epsilon\).
MSC:
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |
| 45F15 | Systems of singular linear integral equations |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 74F15 | Electromagnetic effects in solid mechanics |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 35C20 | Asymptotic expansions of solutions to PDEs |
Keywords:
quasi-linear heat transmission problem; singularly perturbed domain; periodic two-phase dilute composite; asymptotic behavior; real analytic continuation in Banach spaceReferences:
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