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Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials. (English) Zbl 0958.80003
Summary: In this work we analyse a model for radiative heat transfer in materials that are conductive, grey and semitransparent. Such materials are for example glass, silicon, water and several gases. The most important feature of the model is the nonlocal interaction due to exchange of radiation. This, together with nonlinearity arising from the well-known Stefan-Boltzmann law, makes the resulting heat equation nonmonotone. By analysing the terms related to heat radiation we prove that the operator defining the problem is pseudomonotone. Hence, we can prove the existence of a weak solution in the cases where coercivity can be obtained. In the general case, we prove the solvability of the system using the technique of sub- and supersolutions.

##### MSC:
 80A20 Heat and mass transfer, heat flow (MSC2010) 45K05 Integro-partial differential equations 47G10 Integral operators 45R05 Random integral equations