Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side

Abstract

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.