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Potential Estimates and Quasilinear Parabolic Equations with Measure Data

About this Title

Quoc-Hung Nguyen

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 291, Number 1449
ISBNs: 978-1-4704-6722-7 (print); 978-1-4704-7682-3 (online)
DOI: https://doi.org/10.1090/memo/1449
Published electronically: November 8, 2023
Keywords: Quasilinear parabolic equations, renormalized solutions, Wolff parabolic potential, Riesz parabolic potential, Bessel parabolic potential, maximal potential, heat kernel, Radon measures, uniformly thick domain, Reifenberg flat domain, decay estimates, Lorentz spaces, Riccati type equations, capacity

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Table of Contents

Chapters

  • 1. Introduction and main results
  • 2. Nonlinear potential theory to parabolic equations
  • 3. Global gradient estimates for parabolic equations
  • 4. Quasilinear Lane–Emden type and quasilinear Riccati type parabolic equations
  • 5. Appendix

Abstract

In this memoir, we study the existence and regularity of the quasilinear parabolic equations: \begin{equation*} u_t-\operatorname {div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu , \end{equation*} in either $\mathbb {R}^{N+1}$ or $\mathbb {R}^N\times (0,\infty )$ or on a bounded domain $\Omega \times (0,T)\subset \mathbb {R}^{N+1}$ where $N\geq 2$. We shall assume that the nonlinearity $A$ fulfills standard growth conditions, the function $B$ is a continuous and $\mu$ is a radon measure. Our first task is to establish the existence results with $B(u,\nabla u)=\pm |u|^{q-1}u$, for $q>1$. We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with $B\equiv 0$, under minimal conditions on the boundary of domain and on nonlinearity $A$. Finally, due to these estimates, we solve the existence problems with $B(u,\nabla u)=|\nabla u|^q$ for $q>1$.

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