Fractional heat equations with subcritical absorption having a measure as initial data

Abstract

We study existence and uniqueness of weak solutions to (F) partial derivative(t)u+(-Delta)(alpha)u+h(t, u) = 0 in (0,infinity) xR(N), with initial condition u(0, center dot) -nu in R-N, where N >= 2, the operator (-Delta)(alpha) is the fractional Laplacian with alpha is an element of (0, 1), nu is a bounded Radon measure and h : (0,infinity) xR -> R is a continuous function satisfying a subcritical integrability condition.

In particular, if h(t, u) = t(beta)u(p) with beta > -1 and 0 < p < p(beta)* := 1 + 2 alpha(1+beta)/N, we prove that there exists a unique weak solution u(k) to (F) with nu = k delta(0), where delta(0) is the Dirac mass at the origin. We obtain that u(k) -> infinity in (0,infinity) x R-N as k ->infinity for p is an element of 8. (0, 1] and the limit of u(k) exists as k -> infinity when 1 < p < p(beta)*, we denote it by u(infinity). When 1 + 2 alpha(1+beta)/N+2 alpha := p(beta)** < p < p(beta)*, u(infinity) is the minimal self-similar solution of (F)(infinity)partial derivative(t)u+(-Delta)(alpha)u+t(beta)u(p) = 0 in (0,infinity) xR(N) with the initial condition u(0, center dot) = 0 in R-N \{0} and it satisfies u(infinity)(0, x) = 0 for x not equal 0. While if 1 < p < p(beta)**, then u(infinity) = U-p, where U-p is the maximal solution of the differential equation y' + t(beta)y(p) = 0 on R+.