Geodesic
Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side
Applications of Mathematics, Tome 55 (2010) no. 2, pp. 111-149.

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We consider a model for transient conductive-radiative heat transfer in grey materials. Since the domain contains an enclosed cavity, nonlocal radiation boundary conditions for the conductive heat-flux are taken into account. We generalize known existence and uniqueness results to the practically relevant case of lower integrable heat-sources, and of nonsmooth interfaces. We obtain energy estimates that involve only the $L^p$ norm of the heat sources for exponents $p$ close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell's equations or to the Navier-Stokes equations (dissipative heating), with many applications such as crystal growth.
DOI : 10.1007/s10492-010-0005-9
Classification : 35D05, 35D30, 35K05, 35K15, 35K20, 35K55, 35K59, 80A20
Keywords: radiative heat transfer; nonlinear parabolic equation; nonlocal boundary condition; right-hand side in $L^1$ 
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     title = {Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side},
     journal = {Applications of Mathematics},
     pages = {111--149},
     publisher = {mathdoc},
     volume = {55},
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Druet, Pierre-Etienne. Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side. Applications of Mathematics, Tome 55 (2010) no. 2, pp. 111-149. doi : 10.1007/s10492-010-0005-9. http://geodesic.mathdoc.fr/articles/10.1007/s10492-010-0005-9/

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