arXiv subjects

Guojie Zheng

Publications and source records attributed to Guojie Zheng.

4 recordsLinked to original sources

arXiv2026

Quantitative Weak Unique Continuation on Annular Domains for Backward Degenerate Parabolic Equations with Degenerate Interior Points

In this paper, we establish a quantitative weak unique continuation theorem on an annular domain for a backward degenerate parabolic equation with a degenerate interior point. Our methodology hinges on approximating the solution of the degenerate parabolic equation through solutions of non-degenerate parabolic counterparts. Subsequently, we establish Carleman estimates for the non-degenerate parabolic equation across two separate domains. By virtue of these estimates, we deduce a quantitative weak unique continuation property for the degenerate parabolic equation, thereby substantiating the weak unique continuation result for the original degenerate parabolic equation.

math.AP↗

arXiv2025

Observation estimates for a semilinear heat equation in \mathbb{R}^n

This paper studies the state observation problems for the semilinear heat equation in R^n. We derive observation estimates for the equation using the logarithmic convexity property of the frequency function (see [12]). As an application, we show that if two solutions coincide on a nonempty open subset ω\subsetΩat some time T>0, then they must be identical.

math.AP↗

arXiv2021

Unique continuation inequalities for the parabolic-elliptic chemotaxis system

This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain. This system is a simplified version of the chemotaxis model introduced by Keller and Segel. With the aid of priori L^infty-estimates (for solutions of the system) built up in this paper, we treat the semi-linear parabolic equation in the system as a linear parabolic equation, and then use the frequency function method and the localization technique to build up two unique continuation inequalities for the system. As a consequence of the above-mentioned two inequalities, we have the following qualitative unique continuation property: if one component of a solution vanishes in a nonempty open subset at some time T>0, then the solution is identically zero.

math.AP↗

arXiv2020

A unique continuation property for a class of parabolic differential inequalities in a bounded domain

This article is concerned with the unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $Ω$ prescribed with some regularity and growth conditions. Our result shows that the value of the solutions can be determined uniquely by its value on an arbitrary open subset $ω$ in $Ω$ at any given positive time $T$. We also derive the quantitative nature of this unique continuation, that is, the estimate of a $L^2(Ω)$ norm of the initial data on $Ω$, which is majorized by that of solution on the bounded open subset $ω$ at terminal moment $t = T$.

math.OC↗