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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.

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Correspondence to Pierre-Etienne Druet.

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Druet, PE. Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side. Appl Math 55, 111–149 (2010). https://doi.org/10.1007/s10492-010-0005-9

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Keywords

  • radiative heat transfer
  • nonlinear parabolic equation
  • nonlocal boundary condition
  • right-hand side in L 1

MSC 2010

  • 35D05
  • 35K05
  • 35K15
  • 35K55