$L^q$-Extensions of $L^p$-spaces by fractional diffusion equations

Der-Chen Chang; Jie Xiao

doi:10.3934/dcds.2015.35.1905

Article Contents

2015, Volume 35, Issue 5: 1905-1920. Doi: 10.3934/dcds.2015.35.1905

This issue Previous Article Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces Next Article Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem

$L^q$-Extensions of $L^p$-spaces by fractional diffusion equations

Der-Chen Chang^1, and
Jie Xiao^2,

1.
Department of Mathematics and Department of Computer Science, Georgetown University, Washington D.C. 20057
2.
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7

Received: May 2014

Revised: September 2014

Published: May 2017

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Abstract

Based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity defined in the upper-half Euclidean space, this note characterizes a nonnegative Randon measure $\mu$ on $\mathbb R^{1+n}_+$ such that the extension $R_\alpha L^p(\mathbb R^n)\subseteq L^q(\mathbb R^{1+n}_+,\mu)$ holds for $(\alpha,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$ where $u=R_\alpha f$ is the weak solution of the fractional diffusion equation $(\partial_t + (-\Delta_x)^\alpha)u(t, x) = 0$ in $\mathbb R^{1+n}_+$ subject to $u(0,x)=f(x)$ in $\mathbb R^n$.

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Mathematics Subject Classification: Primary: 31C15, 35K08, 35K91.

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