NONUNIQUENESS OF SOLUTIONS OF NONLINEAR HEAT-EQUATIONS OF FAST DIFFUSION-TYPE

We study the existence of infinitely many solutions for the Cauchy pro blem associated with the nonlinear heat equation u(t) = (u(m-1)u(x))(x ) in the fast diffusion range of exponents -1 < m less than or equal t o 0 with initial data u(0) greater than or equal to 0, u(0) not equiva lent to 0. The issue of non-uniqueness arises because of the singular character of the diffusivity for u approximate to 0 . The precise ques tion we want to clarify is: can we have multiple solutions even for in itial data which are far away from the singular level u = 0, for insta nce for u(0) (x) = 1? The answer is, rather surprisingly, yes. Indeed, there are infinitely many solutions for every given initial function. These properties differ strongly from other usual types of heat equat ions, linear or nonlinear. We take as initial data an arbitrary functi on in L(loc)(1), (R), We prove that when the initial data have infinit e integral on a side, say at x = infinity, then we can choose either t o have infinite mass for all small times at least on that side, and th e choice is then unique, or finite mass, and then we need to prescribe a flux function with diverging integral at t = 0, being otherwise qui te general. Moreover, a new parameter appears in the solution set. The behaviour on both ends, x = infinity and x = -infinity is similar and independent.