Fast diffusion equation: uniqueness of solutions with a moving singularity

TL;DR

Addresses the uniqueness of distributional solutions to the fast diffusion equation $u_t=\Delta u^m+f(x,t)$ with a moving Dirac source along a prescribed curve in the supercritical regime $m>m_c=(n-2)/n$. The proof uses a Kato-type inequality and Oleinik-type test functions, localizes near the moving singularity with cutoff functions, and shows error terms vanish, yielding a differential inequality for $f(t)=\int\varphi|u_1-u_2|\,dx$ that forces equality of the two solutions. The main contribution is a rigorous uniqueness result for moving-singularity solutions, extending prior work to a Dirac source traveling along a curve and providing a distributional solution framework. This advances understanding of singular sources in nonlinear diffusion models and links to broader results on fast diffusion equations.

Abstract

We focus on open questions regarding the uniqueness of distributional solutions of the fast diffusion equation (FDE) with a given source term. When the source is sufficiently smooth, the uniqueness follows from standard results. Assuming that the source term is a measure, the existence of different classes of solutions is known, but in many cases, their uniqueness is an open problem. In our work, we focus on the supercritical FDE and prove the uniqueness of distributional solutions with a Dirac source term that moves along a prescribed curve.

Theorems & Definitions (1)

• proof