Initial trace of positive solutions to fractional diffusion equation with absorption

TL;DR

This paper establishes the existence of initial traces for positive solutions to a fractional diffusion equation with absorption, characterizing regular and singular sets, and constructs solutions with prescribed initial traces.

Contribution

It introduces a framework for initial trace analysis of fractional diffusion equations with absorption, including the reverse problem of constructing solutions with given initial data.

Findings

Defined regular and singular initial trace sets.

Proved existence of initial traces for positive solutions.

Developed construction methods for solutions with prescribed initial traces.

Abstract

In this paper, we prove the existence of an initial trace T u of any positive solution u of the semilinear fractional diffusion equation (H) $\partial$ t u + (--$\Delta$) $\alpha$ u + f (t, x, u) = 0 in R * + $\times$ R N , where N $\ge$ 1 where the operator (--$\Delta$) $\alpha$ with $\alpha$ $\in$ (0, 1) is the fractional Laplacian and f : R + $\times$ R N $\times$ R + $\rightarrow$ R is a Caratheodory function satisfying f (t, x, u)u $\ge$ 0 for all (t, x, u) $\in$ R + $\times$ R N $\times$ R +. We define the regular set of the trace T u as an open subset of R u $\subset$ R N carrying a nonnegative Radon measive $\nu$ u such that lim t$\rightarrow$0 Ru u(t, x)$\zeta$(x)dx = Ru $\zeta$d$\nu$ $\forall$$\zeta$ $\in$ C 2 0 (R u), and the singular set S u = R N \ R u as the set points a such that lim sup t$\rightarrow$0 B$\rho$(a) u(t, x)dx = $\infty$ $\forall$$\rho$ \> 0. We study the reverse problem of constructing a positive solution to (H) with a given initial trace (S, $\nu$) where S $\subset$ R N is a closed set and $\nu$ is a positive Radon measure on R = R N \ S and develop the case f (t, x, u) = t $\beta$ u p where $\beta$ \> --1 and p \> 1.