Extended solutions for general fast diffusion equations with optimal measure data

Abstract

We study the theory of existence and uniqueness of nonlinear diffusion equations of the form ut = δφ(u) in ℝN × (0,∞), where φ: ℝ+ → ℝ+ is a continuous and increasing function. We focus on the fast diffusion type by imposing the growth condition, for some constants 0 <1 and all s > 0. Moreover, m1 > (N-2)/N. Existence is obtained for an optimal class of initial data, namely, for any nonnegative Borel measure (not necessarily a locally finite measure). Uniqueness is proven for concave φ's. The results extend to the general equation the optimal theory with measure-valued data now available for the equation with power functions φ(s) = sm. The asymptotic behaviour is also studied.