On variable Lebesgue spaces and generalized nonlinear heat equations

2024/04/15 by Vergara-Hermosilla, Gastón · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.09588

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(⋅)(ℝⁿ). 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 L^p(⋅) _{(nb)/(2α- ⟨ 1 ⟩γ)} (ℝⁿ,L^∞([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(⋅) ( [0,T], L^q (ℝ³) ).

Cited by

• The Fourier transform in variable exponent Lebesgue spaces

Related

• Remarks on variable Lebesgue spaces and fractional Navier-Stokes equations
• On the Cauchy problem for the fractional Keller-Segel system in variable Lebesgue spaces
• Subdifferential calculus and doubly nonlinear evolution equations in L^p spaces with variable exponents
• Heat equation with a nonlinear boundary condition and uniformly local L^r spaces
• On heat equations associated with fractional harmonic oscillators