Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities

Abstract

This paper is devoted to constructing a general theory of nonnegative solutions for the equation

called “the fast-diffusion equation” in the literature. We consider the Cauchy problem taking initial data in the set ℬ⁺ of all nonnegative Borel measures, which forces us to work with singular solutions which are not locally bounded, not even locally integrable. A satisfactory theory can be formulated in this generality in the range 1 > m > m _c = max {(N− 2)/N,0}, in which the limits of classical solutions are also continuous in ℝ^N as extended functions with values in ℝ₊∪{∞}. We introduce a precise class of extended continuous solutions ℰ _c and prove (i) that the initial-value problem is well posed in this class, (ii) that every solution u(x,t) in ℰ _c has an initial trace in ℬ⁺, and (iii) that the solutions in ℰ _c are limits of classical solutions.

Our results settle the well-posedness of two other related problems. On the one hand, they solve the initial-and-boundary-value problem in ?× (0,∞) in the class of large solutions which take the value u=∞ on the lateral boundary x∈∂?, t>0. Well-posedness is established for this problem for m _c < m > 1 when ? is any open subset of ℝ^N and the restriction of the initial data to ? is any locally finite nonnegative measure in ?. On the other hand, by using the special solutions which have the separate-variables form, our results apply to the elliptic problem Δf=f ^q posed in any open set ?. For 1 > q > N/(N− 2)₊ this problem is well posed in the class of large solutions which tend to infinity on the boundary in a strong sense.

As is well known, initial data with such a generality are not allowed for m≧ 1. On the other hand, the present theory fails in several aspects in the subcritical range 0> m≦m _c , where the limits of smooth solutions need not be extended-continuously.