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Existence and uniqueness for renormalized solutions to a general noncoercive nonlinear parabolic equation

T. T. Dang, G. Orlandi

Abstract

This paper introduces the concept of renormalized solution for a general class of non-coercive nonlinear parabolic problems, including both singularities and unbounded lower order terms. We prove existence and uniqueness of renormalized solutions for such class of problems.

Existence and uniqueness for renormalized solutions to a general noncoercive nonlinear parabolic equation

Abstract

This paper introduces the concept of renormalized solution for a general class of non-coercive nonlinear parabolic problems, including both singularities and unbounded lower order terms. We prove existence and uniqueness of renormalized solutions for such class of problems.
Paper Structure (8 sections, 10 theorems, 126 equations)

This paper contains 8 sections, 10 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Lorentz spaces
  3. Existence of renormalized solutions
  4. The approximating problems
  5. A priori estimates
  6. Passing to the limit
  7. Uniqueness of renormalized solutions
  8. Proof of Lemma \ref{['Le: 4.1']} completed

Key Result

Theorem 2.1

FarroniLorentzO'Neil Assume that $1<p<N, 1\le q \le p$. Then, any function $g\in W_0^{1,1}(\Omega)$ satisfying $\vert\nabla g \vert \in L^{p,q}(\Omega)$ belongs to $L^{p^{\star},q}(\Omega)$ where $p^{*}= \frac{Np}{N-p}$ and where $S_{N,p}= \omega_{N}^{-1/N} \frac{p}{N-p}$ and $\omega_{N}$ stands for the measure of the unit ball in $\mathbb{R}^{N}$.

Theorems & Definitions (17)

  • Theorem 2.1
  • Definition 3.1
  • Theorem 3.1: Existence
  • Lemma 3.1
  • proof
  • Lemma 3.2: Lemma 4.1, $Betta$
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 7 more