The heat equation with singular nonlinearity and singular initial data

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Abstract

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut- Δ u = a (x) uq+ b (x) upin a bounded domain and with Dirichlet's condition on the boundary. We consider here a ∈ Lα(Ω), b ∈ Lβ(Ω) and 0 < q ≤ 1 < p. The initial data u (0) = u0is considered in the space Lr(Ω), r ≥ 1. In the main result (0 < q < 1), we assume a, b ≥ 0 a.e. in Ω and we assume that u0≥ γ dΩfor some γ > 0. We find a unique solution in the space C ([0, T], Lr(Ω)) ∩ Lloc∞((0, T), L∞(Ω)). © 2006 Elsevier Inc. All rights reserved.

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Loayza, M. (2006). The heat equation with singular nonlinearity and singular initial data. Journal of Differential Equations, 229(2), 509–528. https://doi.org/10.1016/j.jde.2006.07.007

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