Non-uniqueness for a differential equation and a proof by ChatGPT

TL;DR

This paper investigates conditions for uniqueness in a specific differential equation, demonstrating non-uniqueness for smooth functions and establishing criteria for uniqueness, with insights derived from an AI-generated example.

Contribution

It provides a detailed analysis of non-uniqueness and uniqueness conditions for a class of differential equations, including an example originating from ChatGPT output.

Findings

Existence of smooth solutions with non-zero boundary despite zero initial condition.

Identification of classes of weight functions ensuring uniqueness.

Discussion of AI-generated example and its implications for mathematical proofs.

Abstract

Let $f(t,x),M(t,x)\in C([0,1]^2)$ with $M(t,x)>0$. We consider differential equations of the form \[ \frac{\partial f}{\partial t}(t,x)=\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},\quad x>0. \] For a fixed positive weight $M$, we ask whether the condition $f(0,x)=0$ forces $f\equiv 0$. We show the answer is negative for smooth functions: there exist $f(t,x),M(t,x)\in C^{\infty}([0,1]^2)$ with $f(0,x)=0$, $f(t,0)\not\equiv 0$, and $M(t,x)>0$ satisfying the above equation. However, we show that for a large class of $M(t,x)$, the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro, and we include a discussion of its output as well as a complete proof.