Initial trace of positive solutions to fractional diffusion equation with absorption

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.