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Pré-Publication, Document De Travail Année : 2024

On variable Lebesgue spaces and generalized nonlinear heat equations

Résumé

In this work we address some questions concerning the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces $L^{p(\cdot)}(\R^n)$. More precisely, by mixing some structural properties of these spaces with decay estimates of the fractional heat kernel, we were able to prove two well-posedness results for these equations. In a first theorem, we prove the existence and uniqueness of global-in-time mild solutions in the mixed-space $\mathcal{L}^{p(\cdot)}_{ \frac{nb}{2\alpha - \langle 1 \rangle_\gamma} } (\mathbb{R}^n,L^\infty([0,T[ ))$. On the other hand, by introducing a new class of variable exponents, we demonstrate the existence of an unique local-in-time mild solution in the space $L^{p(\cdot)} \left( [0,T], L^{q} (\R^3) \right)$.
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hal-04545487 , version 1 (14-04-2024)

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  • HAL Id : hal-04545487 , version 1

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Gastón Vergara-Hermosilla. On variable Lebesgue spaces and generalized nonlinear heat equations. 2024. ⟨hal-04545487⟩
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