Non-uniqueness of solutions of nonlinear heat equations of fast diffusion type

Abstract

We study the existence of infinitely many solutions for the Cauchy problem associated with the nonlinear heat equation ut = (um − 1ux)x in the fast diffusion range of exponents −1 <m ≤ 0 with initial data u0 ≥ 0, u0 ≢ 0. The issue of non-uniqueness arises because of the singular character of the diffusivity for u ≈ 0. The precise question we want to clarify is: can we have multiple solutions even for initial data which are far away from the singular level u = 0, for instance for u0 (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 equations, linear or nonlinear.We take as initial data an arbitrary function in Lloc1(R). We prove that when the initial data have infinite integral on a side, say at x = ∞, then we can choose either to have infinite mass for all small times at least on that side, and the choice is then unique, or finite mass, and then we need to prescribe a flux function with diverging integral at t = 0, being otherwise quite general. Moreover, a new parameter appears in the solution set. The behaviour on both ends, x = ∞ and x = −∞ is similar and independent.