Similarity solutions of the porous medium equation with sign changes
Josephus Hulshof · Journal of Mathematical Analysis and Applications · 1991
with m > 1. Here u is a function of x and t. For nonnegative u Eq. (1.1) arises in the theory of gas flow through a one-dimensional porous medium, and an extensive theory has been developed over the last decades (see, e.g., [l, 3, 151). More recently [4, 5, 12, 131 applied and pure mathematicians have also become interested in solutions of (1.1) with sign changes. The most striking property of (1.1) is that solutions may have compact support whose (free) boundaries (or interfaces) move outwards with a finite, possibly zero speed proportional to the one-sided x-derivative of 1~1”~ ’ at the free boundary, taken from the interior of the support. This is usually called the interface condition. The expression between brackets in (l.l), i.e., 1~1~~’ uX= ((l/m) Iu)“--’ u),, which we shall think of as theflux because of the divergence form of (l.l), is continuous with respect to x in zeros of U. If it is nonzero we shall call this the sign change condition. We observe that for u bounded away from zero and infinity Eq. (1.1) is uniformly parabolic and local smoothness follows from standard regularity theory. For the development of the theory of (1.1) similarity solutions play an important role. Essentially these are solutions whose profiles remain the same as t varies. In the nonnegative case they were studied extensively in [7-91. The most familiar of them are of course the Burenblatt-Pattle solutions, given by