arXiv subjects

Kaiqiang Zhang

Publications and source records attributed to Kaiqiang Zhang.

7 recordsLinked to original sources

arXiv2026

A Refined Sum-Product Estimate via Higher Energies

Let $A\subset\mathbb{R}$ be a finite set. Combining the multiplicative slope estimate of Rudnev--Stevens, Cushman's higher-energy regularization, Shakan's $d^+$--$d^\times$ decomposition, and Solymosi's classical sum--product estimate, we prove \[ |AA|^{204}|A+A|^{301}\gtrsim |A|^{675}, \] where $\gtrsim$ suppresses a fixed polylogarithmic factor in $|A|$. Consequently, for every $\varepsilon>0$, \[ \max\{|A+A|,|AA|\}\gg_\varepsilon |A|^{135/101-\varepsilon}. \] The proof is organized around two intermediate estimates. For every nonempty finite set $B\subset\mathbb{R}_{>0}$, \[ d^\times(B)|BB|^{12}|B+B|^{16}\gtrsim |B|^{38}, \] whereas for every nonempty finite set $U\subset\mathbb{R}$, \[ d^+(U)^{17}|U+U|^{29}\gtrsim |U|^{46}. \]

math.CO↗

arXiv2026

Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions

This paper studies the spectral properties and nonlinear asymptotic stability of a family of finite-time self-similar blow-up solutions to the three-dimensional Keller--Segel system constructed by matching interior and exterior profiles within the framework of matched asymptotic expansions. For every sufficiently large matching index $n$, the full linearized operator around the stationary state $U_n$ in self-similar variables is analyzed on $L^2(\mathbb R^3)$. Sturm zero counting in the radial mode, a wave operator that reduces the nonlocal $l=1$ equation to a local equation, and a Mellin--Newton quadratic form for all $l\ge2$ show that, after the scaling and translation modes and the finitely many genuinely unstable radial modes are removed, the remaining spectrum is separated from the imaginary axis by a positive distance. In addition, the optimal lower bound on the real parts of the spectrum is $1/4$ in every mode $l\ge2$. On the stable subspace, an exponentially decaying semigroup and a modified energy equivalent to the $L^2$ norm are constructed, and the logarithmic asymptotic decay rate of the semigroup norm is proved to equal the stable spectral gap. Finally, modulation equations, $H^2$ energy estimates, control of the scaling derivative, and Brouwer's no-retraction theorem yield nonradial nonlinear asymptotic stability of the corresponding self-similar blow-up solutions after the initial coefficients in the finitely many unstable radial directions have been chosen suitably.

math.AP↗

arXiv2026

Finite time blow-up for an inhomogeneous parabolic equation

We consider the inhomogeneous nonlinear heat equation \[ \partial_t u-Δu=|u|^{p-1}u+f(x),\qquad x\in\mathbb{R}^3,\quad p>5, \] where $f\in L^\infty\cap C^{0,1}(\mathbb{R}^3)$. For every sufficiently large integer $n$, we construct a codimension-$n$ Lipschitz manifold of non-radial initial data whose corresponding solutions blow up in finite time and whose rescaled profiles converge to the prescribed self-similar profile $Φ_n$ of the homogeneous equation. The main novelty is to show that the finite-codimensional stability mechanism for self-similar blow-up, developed in the work of Collot, Raphaël and Szeftel [Mem. Amer. Math. Soc. (2019)] for the homogeneous equation, is robust under the addition of a bounded, Lipschitz spatially inhomogeneous source term. In contrast with the homogeneous problem, the equation considered here has no exact scaling invariance, which is a key ingredient in many previous constructions. We expect that the framework developed in this work may also be useful for related problems in which exact scaling invariance is broken.

math.AP↗

arXiv2026

On the existence and nonexistence of global solutions of the semilinear heat equation

We consider the semilinear heat equation $$ u_t-Δu=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling invariance which leads to a potential well of zero depth. We employ the forward similarity transform to convert the equation into a new parabolic equation, so that we can apply the potential well method in weighted Sobolev spaces. As a result, we obtain a new criterion that establishes whether solutions to the heat equation blow up in finite time or exist globally. This work extends the partial results of Ikehata et al. (\textit{Ann. Inst. H. Poincaré Anal. Non Linéaire}, \textbf{27} (2010) 877-900) from critical Sobolev exponent to the case $p_F<p<p_S$, where $p_F=1+2/n$ is the Fujita exponent and $p_S=(n+2)/(n-2)$ (for $n\ge3$) is the critical Sobolev exponent.

math.AP↗

arXiv2025

Infinitely many self-similar blow-up profiles for the Keller-Segel system in dimensions 3 to 9

Based on the method of matched asymptotic expansions and Banach fixed point theorem, we rigorously construct infinitely many self-similar blow-up profiles for the parabolic-elliptic Keller-Segel system \begin{equation*} \left\{\begin{array}{l} \partial_{t} u=Δu-\nabla \cdot\left(u \nabla Φ_{u}\right), \\ 0=ΔΦ_{u}+u,\\ u(\cdot,0)=u_0 \geq 0 \end{array}\quad \text{in}\ \mathbb{R}^{d},\right. \end{equation*} where $d\in \{3,\cdots,9\}$. Our findings demonstrate that the infinitely many backward self-similar profiles approximate the rescaling radial steady-state near the origin (i.e. $0<|x|\ll1$) and $\frac{2(d-2)}{|x|^2}$ at spatial infinity (i.e. $|x|\gg1$). We also establish the convergence of the self-similar blow-up solutions as time tends to the blow-up time $T>0$. Our results can give a refined description of backward self-similar profiles for all $|x|\geq 0$ rather than for $0<|x|\ll1$ or $|x|\gg1$, indicating that the blow-up point is the origin and $$ u(x,t)\sim \frac{1}{|x|^2},\ \ \ x\ne0,\ \text{as}\ t\to T. $$

math.AP↗

arXiv2024

On the stability of Type I self-similar blowups for the Keller-Segel system in three dimensions and higher

We consider the parabolic-elliptic Keller-Segel system in spatial dimensions $d\geq3$, which corresponds to the mass supercritical case. Some solutions become singular in finite time, an important example being backward self-similar solutions. Herrero et al. and Brenner et al. showed the existence of such profiles, countably many in dimensions $3\leq d \leq 9$ and at least two for $d\geq 10$. We establish that all these self-similar profiles are stable along a set of initial data with finite Lipschitz codimension equal to the number of instable eigenmodes. This extends the recent finding of Glogić et al. showing the stability of the fundamental self-similar profile. We obtain additional results, such as the possibility of the solutions we construct to originate from smooth and compactly supported initial data, their convergence at blow-up time, and the Lipschitz regularity of the blow-up time. Our proof extends the approach proposed in Collot et al., based on renormalizing the solution around a modulated self-similar solution, and using a spectral gap for the linearized operator in the parabolic neighbourhood of the singularity.

math.AP↗

arXiv2022

Set-theoretical solutions to the Hom-Yang-Baxter equation and Hom-cycle sets

Set-theoretic solutions to the Yang-Baxter equation have been studied extensively by means of related algebraic systems such as cycle sets and braces, dynamical versions of which have also been developed. No work focuses on set-theoretic solutions to the Hom-Yang-Baxter equation (HYBE for short). This paper investigates set-theoretic solutions to HYBE and associated algebraic system, called Hom-cycle sets. We characterize left non-degenerate involutive set-theoretic solutions to HYBE and Hom-cycle sets, and establish their relations. We discuss connections among Hom-cycle sets, cycle sets, left non-degenerate involutive set-theoretic solutions to HYBE and the Yang-Baxter equation.

math.RA↗