A nonlinear oblique derivative boundary value problem for the heat equation: analogy with the porous medium equation
This paper studies the heat equation in the upper half-plane with diffuison in only one direction together with an oblique derivative boundary condition. Global existence for the Cauchy problem is proved and qualitative properties of the solutions are stated: this means compact support for solutions, finite speed of propagation, free boundary relations and convergence to self-similar solutions as asymptotic behavior. This problem exhibits a strong analogy with the porous media equation. The problem arises in modelling a plasma opening switch. The main result gives existence of a global classical solution for suitable initial data together with a \(C^{1,\alpha}\) free boundary and the smoothness of the ``pressure'' \(u^2\) for the solution \(u\). Existence is proved by an approximation argument which is developed in Section 2, where interesting comparison results are also obtained. Finally, existence of a unique viscosity solution is obtained; moreover, there is a comparison result for viscosity solutions. Abel type integral operators are used here. Section 3 deals with travelling waves and self-similar solutions, and it is proved that the latter are also viscosity solutions. The proof that the interface is Lipschitz and nondegenerate is given in Section 4, and also that similar properties hold for the pressure as well. Some Harnack inequalities are also proved. The \(C^{1,\alpha}\) property of the free boundary is proved in Section 5 by using an iteration technique of Caffarelli-Wolanski and, finally, the asymptotic behavior and the free boundary relations are studied in Section 6.
scientific article; zbMATH DE number 1785832