On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Ping Li; Pablo Raúl Stinga; José L. Torrea

doi:10.3934/cpaa.2017041

Article Contents

2017, Volume 16, Issue 3: 855-882. Doi: 10.3934/cpaa.2017041

This issue Previous Article On nonexistence of solutions to some nonlinear parabolic inequalities Next Article $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

1.
School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
3.
Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, USA

* Corresponding author
* Corresponding author

Received: June 30, 2016

Revised: December 31, 2016

Published: January 2018

The first author was supported by grants 11471251 and 11271293 from National Natural Science Foundation of China. The second and third authors were supported by grant MTM2015-66157-C2-1-P from Government of Spain.

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Abstract

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation

$ \partial_tu=\Delta u+f, $

the harmonic oscillator evolution equation

$\partial_tu=\Delta u-|x|^2u+f, $

and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0 < s < 1$.

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Mathematics Subject Classification: Primary: 35K10, 35B45, 42B37; Secondary: 58J35, 42B20.

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