Inverse problems for some fractional equations with general nonlinearity
Journal
Research in Mathematical Sciences
Journal Volume
10
Journal Issue
4
Date Issued
2023-12-01
Author(s)
Kow, Pu Zhao
Abstract
Inspired by some interesting equations modeling anomalous diffusion and nonlinear phenomena, we will study the inverse problems of uniquely identifying coefficients in nonlinear terms from over-determined data. Precisely, we consider a semilinear fractional Schrödinger operator (- Δ) su+ Q(x, u) = 0 in Ω with 0 < s< 1 . The fractional Laplacian arises due to the anomalous diffusion, e.g., the motion of particles described by Lévy flights. Here, we consider the semilinear term Q(x, u) = q(x, | u|) u , which appears naturally in the study of nonlinear optics with cubic Kerr-type nonlinearity, the complex Ginzburg–Landau equation with cubic-quintic nonlinearity, or even the Hartree equation with convolution-type nonlinearity, etc. In this article, we consider the time-independent and the time-evolution semilinear fractional Schrödinger equations with “Dirichlet” condition given on the complement of Ω . We prove both the well-posedness of the forward problems and the unique determination of the inverse problems with measurements taken on the complement of Ω .
Subjects
Fractional Laplacian | Global uniqueness of inverse problems | Hartree potentials | Kerr-type nonlinearity | Nonlinear potentials
SDGs
Type
journal article