An existence and uniqueness result to evolution equations with sign-changing pseudo-differential operators and its applications to logarithmic Laplacian operators and second-order differential operators without ellipticity

2024/03/29 by Jae-Hwan Choi, Choi, Jae-Hwan, Ildoo Kim +1
Computer Science · Mathematics · #35A01 #35D30 #35S05 #35S10 #47G30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2403.19968

openalex publication_date 2024/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator log (-Δ) and a second-order differential operator whose leading coefficients are not positive semi-definite.

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