Uniqueness/nonuniqueness for nonnegative solutions of the Cauchy problem for $u_t=Δu-u^p$ in a punctured space

分数抛物型方程的非唯一性：亚线性幂型非线性情况

Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity

本文研究了分数抛物型方程在亚线性幂型非线性情况下的非唯一性，证明了在特定条件下存在非平凡解，与超线性情况下的唯一零解形成对比。

Nonuniqueness in Vector-Valued Calculus of Variations in $L^\infty$ and some Linear Elliptic Systems

For a Hamiltonian $H \in C^2(\mathbb{R}^{N \times n})$ and a map $u:Ω\subseteq \mathbb{R}^n /!\longrightarrow \mathbb{R}^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,Ω) \ :=\ \big\|H(Du)\big\|_{L^\infty(Ω)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system \[ \label{2} \tag{2} A_\infty u := \Big(H_P \otimes H_P + H[H_P]^\bot /! H_{PP}\Big)(Du):D^2 u = 0. \] \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author [K1]. Herein we show that the D…

具有Sobolev速度场的输运方程中的尖锐非唯一性

Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

本文通过构造反例证明DiPerna--Lions理论中输运方程弱解唯一性条件在Sobolev速度场下的改进范围是尖锐的，并提出一种兼具凸积分与自相似特征的新颖流动机制。

Isolated Singularities of Polyharmonic Inequalities

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-Δ^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these solutions $u$ satisfy a pointwise a priori bound as $x\to 0$.