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On variable Lebesgue spaces and generalized nonlinear heat equations (2404.09588v2)

Published 15 Apr 2024 in math.AP

Abstract: 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 $L{p(\cdot)}(\mathbb{R}n)$. 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 $L{p(\cdot)} \left( [0,T], L{q} (\mathbb{R}3) \right)$.

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