Regularity of interfaces for an inhomogeneous filtration equation (Q2757089)

This paper is concerned with the regularity of the interfaces for nonnegative solutions \(u\) of the Cauchy problem for the one-dimensional degenerate diffusion equation \(\rho(x)u_t=(u^m)_{xx}\) with \(m>1\). The function \(\rho\) is assumed to be even, positive and in \(C^2(\mathbb{R})\). The initial function \(u_0(x)\) is assumed to be even, nonnegative, bounded and continuous, and to have a bounded support \([-a,a]\). As opposed to the classical porous medium equation, it is known that the interfaces in this equation may propagate with infinite speed and disappear with in a finite time provided the function \(\rho\) decreases fast enough to zero as \(|x|\to\infty\) (see \textit{V. A. Galaktionov} and \textit{J. R. King} [IMA J. Appl. Math. 57, 53--77 (1996; Zbl 0869.35076)], \textit{M. Guedda}, \textit{D. Hilhorst} and \textit{M. A. Peletier} [Adv. Math. Sci. Appl. 7, 695--710 (1997; Zbl 0891.35071)], \textit{S. Kamin} and \textit{R. Kersner} [Meccanica 28, 117--120 (1993; Zbl 0786.76088)], and \textit{M. A. Peletier} [Appl. Math. Lett. 7, 29--32 (1994; Zbl 0803.35172)].NEWLINENEWLINE In the present paper the authors establish conditions on \(\rho\) and \(u_0\) sufficient to provide the local \(C^1\)-regularity of the interfaces until the moment of their disappearance, and obtain the interface equation (Darcy's law). Since the equation with \(\rho(x)\) is inhomogeneous, Lagrangian coordinates are used and the problem is transformed into a problem for a nonlocal parabolic equation posed in a fixed bounded domain.