Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and right-hand side in $L^p$ with $pge 1$
- Druet, Pierre-Étienne
2010 Mathematics Subject Classification
- 35D05 35K05 35K15 35K55
- Nonlinear parabolic equation, nonlocal boundary condition, right-hand side in L-p with p>=1
It is known that the time-dependent heat equation with nonlocal radiation boundary conditions possesses a unique weak solution if the heat sources are in L-2. In this paper, we generalize the known existence and uniqueness results to the case that the right-hand side belongs to an arbitrary L-p space (p >= 1). This is the continuation of the results that we recently proved for the stationary problem. The purpose of both papers is to obtain energy estimates that involve only the L-p norm of the heat sources for some exponent 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).
- Appl. Math., 55 (2010) pp. 111--149.