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) ∂ t u + (−∆) α u + f (t, x, u) = 0 in R * + $\times$ R N , where N ≥ 1 where the operator (−∆) α with α ∈ (0, 1) is the fractional Laplacian and f : R + $\times$ R N $\times$ R + → R is a Caratheodory function satisfying f (t, x, u)u ≥ 0 for all (t, x, u) ∈ R + $\times$ R N $\times$ R +. We define the regular set of the trace T u as an open subset of R u ⊂ R N carrying a nonnegative Radon measive ν u such that lim t→0 Ru u(t, x)ζ(x)dx = Ru ζdν ∀ζ ∈ 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→0 Bρ(a) u(t, x)dx = ∞ ∀ρ > 0. We study the reverse problem of constructing a positive solution to (H) with a given initial trace (S, ν) where S ⊂ R N is a closed set and ν is a positive Radon measure on R = R N \ S and develop the case f (t, x, u) = t β u p where β > −1 and p > 1.
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Submitted on : Tuesday, October 23, 2018 - 2:14:11 PM
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  • HAL Id : hal-01662134, version 4
  • ARXIV : 1712.05223

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Huyuan Chen, Laurent Veron. Initial trace of positive solutions to fractional diffusion equation with absorption. Journal of Functional Analysis, Elsevier, 2019, 276, pp.1145-1200. ⟨https://doi.org/10.1016/j.jfa.2018.10.013⟩. ⟨hal-01662134v4⟩

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