Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities

This paper is devoted to constructing a general theory of nonnegative solutions for the equation u(t) = Delta(u(m)), 0<m<1, 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+ U {infinity}. 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 R x (0, infinity) in the class of large solutions which take the value u=infinity on the lateral boundary x is an element of partial derivativeR, t>0. Well-posedness is established for this problem for m(c)<m<1 when R is any open subset of R-N and the restriction of the initial data to R is any locally finite nonnegative measure in R. On the other hand, by using the special solutions which have the separate-variables form. our results apply to the elliptic problem Deltaf=f(q) posed in any open set R. 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 mgreater than or equal to1, 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.