Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators

Abdelhamid Gouasmia, Hichem Hajaiej, Kaushik Bal

Abstract

In this work, we address a parabolic problem featuring a potentially doubly nonlinear term, governed by a combination of local and nonlocal operators (see Problem P1 below). We first establish the local existence of weak energy solutions via a semidiscretization in time applied to an auxiliary evolution problem. The uniqueness of these solutions is subsequently obtained through a novel generalization of the classical inequality of Diaz and Saa, suitably adapted to the mixed local nonlocal setting. This generalization provides a new comparison principle and establishes the T-accretivity of a corresponding operator in L2. By employing this comparison principle, we construct suitable barrier functions that allow the global in time extension of solutions. Furthermore, we demonstrate the convergence of weak solutions to a nontrivial stationary state. Our approach relies on methods from the theory of contraction semigroups. It is noteworthy that these results are underpinned by a detailed analysis of the stationary problems associated with Problem P1, which also reveals several qualitative properties of the solutions.

Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators

Abstract

In this work, we address a parabolic problem featuring a potentially doubly nonlinear term, governed by a combination of local and nonlocal operators (see Problem P1 below). We first establish the local existence of weak energy solutions via a semidiscretization in time applied to an auxiliary evolution problem. The uniqueness of these solutions is subsequently obtained through a novel generalization of the classical inequality of Diaz and Saa, suitably adapted to the mixed local nonlocal setting. This generalization provides a new comparison principle and establishes the T-accretivity of a corresponding operator in L2. By employing this comparison principle, we construct suitable barrier functions that allow the global in time extension of solutions. Furthermore, we demonstrate the convergence of weak solutions to a nontrivial stationary state. Our approach relies on methods from the theory of contraction semigroups. It is noteworthy that these results are underpinned by a detailed analysis of the stationary problems associated with Problem P1, which also reveals several qualitative properties of the solutions.

Paper Structure

This paper contains 12 sections, 17 theorems, 373 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Main results for the stationary problems related to problem \ref{['PP1']}
  4. Main Results for Problem \ref{['P1']}
  5. Stationary Problems Associated with Problem \ref{['PP1']}
  6. Weak Comparison Principle for Problem \ref{['PP2']}
  7. Existence and Uniqueness of Weak Solutions for Problem \ref{['PP2']}
  8. Existence and Uniqueness of Weak Solutions and Accretivity Results for Problem \ref{['equ22']}
  9. Existence and Uniqueness of Weak Solutions for Problem \ref{['P7']}
  10. Existence, Uniqueness, Regularity, and Asymptotic Analysis of problem \ref{['P1']}
  11. Existence, Uniqueness, and Regularity of Problem \ref{['P1']}
  12. Asymptotic Analysis of Problem \ref{['P1']}

Key Result

Theorem 2.1

Let $p \geq 1$ with $N > p$. Then there exists a positive constant $C = C(N, p, \Omega)$ such that, for every function $u \in W^{1, p}_{0}(\Omega)$, the following inequality holds: where $p^{*}$ denotes the Sobolev critical exponent, given by $p^{*} = \frac{Np}{N - p}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (41)

  • Theorem 2.1: evans2022partial
  • Remark 2.1
  • Theorem 2.2: di2012hitchhikers
  • Remark 2.2
  • Remark 2.3
  • Definition 2.1
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 31 more