Uma equação do calor semilinear com uma reação localizada e dados iniciais não contínuos

Abstract

This dissertation studies a semilinear heat equation with a nonlinear term repre senting a concentrated reaction in a closed region of Ω ⊂ Rn, a smooth and bounded domain, or the entire space Rn, with n ≥ 2. The work is based on the article [9], which allows for the consideration of non-continuous initial data. We first present the preliminary concepts and analytical tools required for the development of the theory, including special functions, results from measure theory, and Lebesgue spaces Lr(Ω), with 1 < r < ∞. We then study the linear heat equation and the associated Green’s function, deriving fundamental linear estimates in both the interior of the domain and on the boundary, which are essential for handling the localized nonlinear term. Finally, we establish the main result of the dissertation, proving local-in time existence, uniqueness of solutions, continuous dependence on the initial data, and preservation of positivity by means of Banach’s Fixed Point Theorem.