A fractional porous medium equation

We develop a theory of existence, uniqueness and regularity for the following porous medium equation with fractional diffusion, with m>m_?=(N−1)/N, N?1 and f∈L¹(RN). An L¹-contraction semigroup is constructed and the continuous dependence on data and exponent is established. Nonnegative solutions are proved to be continuous and strictly positive for all x∈RN, t>0.     

Symmetry of the blow-up set of a porous medium type equation

We study the blow-up set of a porous medium type equation with source. Under some technical conditions, we prove that if the blow-up set is a bounded smooth region, then it must be a ball with a certain radius. This problem can be reduced to a sublinear elliptic equation coupled with an overdetermined boundary condition. Roughly speaking, the overdetermined boundary condition forces the domain to be a ball. Because the nonlinear term is sublinear and then non-Lipschitz, many difficulties arise if one wants to use the moving plane method to reach the goal. In particular, the Hopf boundary lemma is not applicable to this problem. Instead, we investigate various related problems in a half space and a problem in the first quadrant of the entire space, and then use the symmetry results obtained for these problems to overcome the obstacles encountered. ©1995 John Wiley & Sons, Inc.     

A remark on the porous medium equation with nonlinear source

In the present paper, we obtain a new a priori estimate of the solution of the initial-boundary value problem for the porous medium equation with nonlinear source and formulate the conditions guaranteeing the global solvability of this problem.     

Multiple blow-up for a porous medium equation with reaction

The present paper is concerned with the Cauchy problem

Lower semicontinuous obstacles for the porous medium equation

We deal with the obstacle problem for the porous medium equation in the slow diffusion regime $m>1$. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.     

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).