Abstract
We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of the viscosity solutions theory, we show that the free boundary uniformly converges to the equilibrium as $t$ grows. In the case of a convex potential, an exponential rate of free boundary convergence is obtained.
Highlights
Consider a C2 function Φ(x) : Rn → R, and consider a nonnegative, continuous function ρ0(x) : Rn → R which has compact support Ω0
We introduce a notion of viscosity solution for the free boundary problem associated with this equation, which we will show to be equivalent to the usual notion of weak solutions – see [7] for the general theory of viscosity solutions
We introduce the appropriate notion of viscosity solution for (PME–D) and show that it is equivalent to the usual notion of weak solution
Summary
Consider a C2 function Φ(x) : Rn → R, and consider a nonnegative, continuous function ρ0(x) : Rn → R which has compact support Ω0. In [11], existence and uniqueness of solutions are established for the full space case under reasonable assumptions (either the initial data is compactly supported or the potential has less than quadratic growth at infinity). Where the first equality is due to the fact that u = 0 on Γ(u) In this regard we closely follow the framework and arguments set out in [6] (see [14] and [5]), where the viscosity concept is introduced and studied for the Porous Medium Equation. In the case of Φ(x) = |x|2 (that is for the renormalized (PME)) Lee and Vazquez [16] showed that the interface becomes convex in finite time It is unknown whether such results hold for general convex potentials: we shall investigate this in an upcoming work
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