Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Renormalized and entropy solutions to the general nonlinear parabolic equations in Musielak-Orlicz spaces

Ying Li, Chao Zhang

Abstract

We study the well-posedness of solutions to the general nonlinear parabolic equations with merely integrable data in time-dependent Musielak-Orlicz spaces. With the help of a density argument, we establish the existence and uniqueness of both renormalized and entropy solutions. Moreover, we conclude that the entropy and renormalized solutions for this equation are equivalent. Our results cover a variety of problems, including those with Orlicz growth, variable exponents, and double-phase growth.

Renormalized and entropy solutions to the general nonlinear parabolic equations in Musielak-Orlicz spaces

Abstract

We study the well-posedness of solutions to the general nonlinear parabolic equations with merely integrable data in time-dependent Musielak-Orlicz spaces. With the help of a density argument, we establish the existence and uniqueness of both renormalized and entropy solutions. Moreover, we conclude that the entropy and renormalized solutions for this equation are equivalent. Our results cover a variety of problems, including those with Orlicz growth, variable exponents, and double-phase growth.

Paper Structure

This paper contains 3 sections, 8 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Functional setting and main tools
  3. The proofs of main results

Key Result

Theorem 1.3

Assume that $f\in L^{1}(\Omega_{T})$, $u_{0}\in L^{1}(\Omega)$, ${\mathcal{A}}$ satisfies the conditions $(A1)$--$(A3)$, $N$-function $M$ is regular enough so that the set of smooth functions is dense in $\textbf{W}(\Omega_T)$ in the modular topology, and $M$ satisfies $\mathsf{(Y)}$-condition. Then

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 6 more