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

TL;DR

This paper proves local existence, uniqueness, regularity, and global behavior of solutions for evolution equations involving mixed local and nonlocal operators, using semidiscretization, comparison principles, and contraction semigroup methods.

Contribution

It introduces a novel comparison principle for mixed local and nonlocal operators, enabling global solution extension and analysis of stationary states.

Findings

Established local existence of weak energy solutions.

Proved uniqueness via a generalized Diaz-Saa inequality.

Demonstrated convergence to stationary states.

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.