Yuri Bozhkov, Igor Leite Freire
We study the Lie point symmetries of semilinear Kohn-Laplace equations on the Heisenberg group H^1 and obtain a complete group classification of these equations.
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We show that any Lie point symmetry of semilinear Kohn-Laplace equations on the Heisenberg group H^1 with power nonlinearity is a divergence symmetry if and only if the corresponding exponent assumes critical value.
Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group carried out in a preceding work, we establish the conservation laws for the critical Kohn-Laplace equations via the Noether's Theorem.
We show which Lie point symmetries of non-critical semilinear Kohn-Laplace equations on the Heisenberg group $H^1$ are Noether symmetries and we establish their respectives conservations laws.
We use a suitable transform related to Sobolev inequality to investigate the sharp constants and optimizers for some Caffarelli-Kohn-Nirenberg-type inequalities which are related to the weighted $p$-Laplace equations. Moreover, we give the classification to the linearized problem related to the radial extremals. As an application, we investigate the gradient type remainder term of related inequality by using spectral estimate combined with a compactness argument which extends the work of Figalli and Zhang (Duke Math. J. 2022) at least for radial case.
We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.
In this manuscript, in the $L^1$ scaling critical case, a lifespan estimate of solutions to the Cauchy problem for non-gauge invariant semilinear semirelativistic equations is considered. The lifespan estimate is given by the modified test function method with a fractional Laplace operator. The main obstacle to obtaining the lifespan estimate is the non-locality of the fractional Laplace operator. To treat the non-locality, special test functions are introduced.
Rigidity of solutions to singular/degenerate semilinear critical equations
本文研究奇异/退化半线性临界方程，证明了正解的刚性结果，并分类了内蕴维数2<n<4时可能无限能量的解。
Liouville theorems for anisotropic $p$-Laplace equations with a semilinear term
针对带半线性项的各向异性 $p$-Laplace 方程，结合 Serrin 技巧与不变张量方法，建立正解、超/子解及稳定变号解的 Liouville 不存在性与分类结果。
Gradient stability of Caffarelli-Kohn-Nirenberg inequality involving weighted p-Laplace
本文研究含加权 p-Laplace 的 CKN 不等式，通过线性化问题分类证明极值元非退化，并结合谱估计与紧性论证建立梯度型余项稳定性。
Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group
本文在海森堡群上研究带强迫项的分数次亚拉普拉斯半线性热方程，确定其 Fujita 指数为 Q/(Q-2s)，并证明超临界情形全局解存在、次临界情形全局解不存在、临界情形存在有限时间爆破解，从而将经典 Fujita 现象推广到次黎曼非局部框架。
A monotonicity formula for a semilinear fractional parabolic equation
针对全分数阶半线性热方程，研究人员通过高维抛物-椭圆变换建立了对应的延拓问题的单调性公式，该公式是局部方程Giga-Kohn单调性公式的分数阶类似物，其证明方法或可推广至其他非线性非局部场景。
Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces
本文针对测度空间上带自伴算子的半线性阻尼波动方程，在热半群衰减估计下建立推广的松村估计，证明小数据全局存在性，并涵盖多种重要算子实例。
We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this Lyapunov functional, we derive the blow-up rate and the blow-up limit of the solution. We also classify all asymptotic behaviors of the solution at the singularity and give precisely blow-up profiles corresponding to these behaviors. Finally, we attain the blow-up profile numerically, thanks to a new mesh-refinement algorithm inspired by the rescaling method of Be…
Global branching for semilinear fractional Laplace with sublinear nonlinearity
本文研究次线性分数阶椭圆问题$(P_λ^s)$正解的存在性、不存在性和多重性，通过先验估计、上下解方法及经典连接定理证明至少存在两个不同正弱解，扩展了已有结果至非局部情形。
We consider the semilinear diffusion equation $\partial$ t u = Au + |u| $α$ u in the half-space R N + := R N --1 x (0, +$\infty$), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ' , 0) = 0 for x ' $\in$ R N --1. We prove that if the symbol of the operator A is of order a|$ξ$| $β$ near the origin $ξ$ = 0, for some …
Entire Nodal Solutions with Prescribed Symmetry to Caffarelli-Kohn-Nirenberg Equations
针对Caffarelli-Kohn-Nirenberg型带权临界p-Laplace方程，本文通过分析对称构型的性质与相容条件，证明了无穷多类指定对称性下非平凡变号整体节点解的存在性。
Starting with approximate solutions of the equation $-Δu=wu^3$ on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a computer, using a Banach algebra of real analytic functions, based on Zernike polynomials. Besides proving existence, and symmetry properties, we also determine the Morse index of the solutions.
On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity
该研究利用变分法与逼近方法，考察兼具局部与非局部 Laplace 算子的奇异半线性椭圆方程，证明参数依赖情形下弱解存在及扰动情形下多解存在。
A Struwe-type Decomposition Result for Weighted Critical $p$-Laplace Equations
本文在包含原点的有界区域中，为Caffarelli-Kohn-Nirenberg型临界p-Laplace方程建立了Palais-Smale序列的Struwe型分解，并引入新的重标度律以处理权函数带来的新困难。
We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the $p$-Laplace operator and a general nonlinearity satisfying concavity type assumptions. This provides an extension of results previously known in the literature only for the torsion and the eigenfunction equations. In the semilinear case $p = 2$ the results are already new since they include new admissible nonlinearities.
We consider inverse problems in space-time $(M, g)$, a $4$-dimensional Lorentzian manifold. For semilinear wave equations $\square_g u + H(x, u) = f$, where $\square_g$ denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map $L: f \rightarrow u|_V$, where $V$ is a neighborhood of a time-like geodesic $μ$, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set where waves can propagate from $μ$ and return back. Moreover, on a given space-time $(M, g)$, the source-to-solution map determines som…
We investigate symmetry properties of solutions to equations of the form $$ -Δu = \frac{a}{|x|^2} u + f(|x|, u)$$ in R^N for $N \geq 4$, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with low Morse index (namely 0 or 1) and which are biradial (i.e. are invariant under the action of a toric group of rotations), are in fact completely radial. A similar result holds for the semilinear Laplace-Beltrami equations on the sphere. Furthermore, we show that the condition on the Morse index is sharp. Finally we apply the result in order to estimate best cons…
Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u = h(x) u^{p-1}$ for $2<p$. The function $h$ may have an indefinite sign. Key ingredients are a $h$-dependent concentration-compactness Lemma and a characterization of compact embeddings of $D^{1,2}(\rz^N)$ into weighted Lebesgue spaces.
Third order estimates and the regularity of the stress field for solutions to $p$-Laplace equations
本文研究了当$p$趋近于半线性极限情形$p=2$时$p$-Laplace方程解的性质，获得了三阶估计，并据此推导出应力场的改进正则性。
Here we design boundary feedback stabilizers to unbounded trajectories, for semi-linear stochastic heat equation with cubic non-linearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. Then, the simple form of the feedback, we propose here, allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed…
Blow-up and global existence for semilinear parabolic equations on infinite graphs
本文研究无限图上半线性热方程，在谱下确界为正的条件下，依据源项在零点附近行为与谱下确界的相互作用，给出了非平凡初值下解全局存在或非负解爆破的判别条件。
Giga-Kohn-type results for the fully fractional heat equation
研究全分数阶热方程，通过厄米特展开对经典泊松扎耶夫恒等式给出新解释，推广了\(\sigma = 1\)时的单调性公式和刘维尔型定理，还建立了时空非局部单调性公式，为相关研究提供了新方法和结果。
We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω, \] where $0 < q < 1$, $σ\ge 0$ is a Radon measure in $Ω$, $ f \ge 0$ is a measurable function, and $\mathbf{G}$ is a linear integral operator with positive kernel $G$ on $Ω\timesΩ$. For quasi-metric (or quasi-metrically modifiable) kernels $G$, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions $u \in L^q_{\rm loc} (Ω, σ)$. Applications are considered to sem…
We verify the critical case $p=p_0(n)$ of Strauss' conjecture (1981) concerning the blow-up of solutions to semilinear wave equations with variable coefficients in $\mathbf{R}^n$, where $n\geq 2$. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when $p=p_0(n)$. The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou(2007) and Zhou and Han (2014): exponential "ei…
We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to $$ {div}\big(\mathsf A\,\nabla v\big)+\mathsf B\,f(v)=0\,,\quad\lim_{|x|\to+\infty}v(x)=0,\quad x\in\mathbb R^n,$$ $n>2$, where $\mathsf A$ and $\mathsf B$ are two positive, radial, smooth functions defined on $\mathbb R^n\setminus\{0\}$. We assume that the nonlinearity $f\in C(-c,c)$, $0<c\le\infty$ is an odd function satisfying some convexity and growth conditions, and has a zero at $b>0$, is non positive and not identically 0 i…
We consider positive critical points of Caffarelli-Kohn-Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted $p$-Laplace operator, which we consider for a general $p \in (1,d)$. For $p=2$, the symmetry breaking region for extremals of Caffarelli-Kohn-Nirenberg inequalities was completely characterized in [J. Dolbeault, M. Esteban, M. Loss, Invent. Math. 44 (2016)]. Our results extend this result to a ge…
We show that the Sobolev embedding is compact on punctured manifolds with conical singularities. On the other hand, we find the Sobolev inequality does not hold on punctured manifolds with Poincaré like metric, on which one has Poincaré inequality. Applying the results to the Laplace's equation on the singular manifolds, we obtain the existences of the solution in both cases. In conical singularity case, we prove further that the solution can be extended to the singular points and it is Hölder continuous. However, the solution can not be continuously extended to the singular points in Poincaré…
Optimal control of a class of semilinear fractional elliptic equations
针对与谱分数阶 Dirichlet Laplace 算子相关的半线性椭圆方程，研究其最优控制问题，证明最优解存在性并给出一阶与二阶最优性条件。
Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones
本文研究了锥内球面扇区中半线性椭圆方程，通过分析径向正解的Morse指数及其对单位球面区域的依赖性，证明了Gidas-Ni-Nirenberg型径向对称性结果在一般非凸锥中不成立，并发现无界锥中标准bubbles解在非凸锥中会变得不稳定。
We consider the Cauchy problem of the semilinear wave equation with a damping term \begin{align*} u_{tt} - Δu + c(t,x) u_t = |u|^p, \quad (t,x)\in (0,\infty)\times \mathbb{R}^N,\quad u(0,x) = \varepsilon u_0(x), \ u_t(0,x) = \varepsilon u_1(x), \quad x\in \mathbb{R}^N, \end{align*} where $p>1$ and the coefficient of the damping term has the form \begin{align*} c(t,x) = a_0 (1+|x|^2)^{-α/2} (1+t)^{-β} \end{align*} with some $a_0 > 0$, $α< 0$, $β\in (-1, 1]$. In particular, we mainly consider the cases $ α< 0, β=0$ or $α< 0, β= 1$, which imply $α+ β< 1$, namely, the damping is spatially increasi…
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem $ -Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $N> 2,$ is a bounded smooth domain, and $f$ is a subcritical Carathéodory non-linearity. We …
We propose a fine analysis of second order optimality conditions for the optimal control of semi-linear parabolic equations with respect to the initial condition. More precisely, we investigate the following problem: maximise with respect to $y\in L^\infty({(0;T)\times Ω})$ the cost functional $J(y)=\iint_{(0;T)\times Ω}j_1(t,x,u)+\int_Ωj_2(x,u(T,\cdot))$ where $\partial_t u-Δu=f(t,x,u)+y\,, u(0,\cdot)=u_0$ with some classical boundary conditions, under constraints of the form $-κ_0\leq y\leq κ_1\text{ a.e.}\,, \int_Ωy(t,\cdot)=V_0$. This class of problems arises in several application fields.…
Sharp pointwise estimates for weighted critical $p$-Laplace equations
本文研究由 Caffarelli–Kohn–Nirenberg 不等式导出的加权拟线性椭圆方程解的渐近行为，利用 Kelvin 型变换获得比非加权情形更精细的无穷远处锐点态估计，并克服了拟线性情形下应用该变换所需的预备估计困难。
Sharp lifespan estimates for semilinear fractional evolution equations with critical nonlinearity
本文研究由分数阶Laplace算子驱动的具临界非线性的半线性波动与二阶σ-演化方程，通过Dini条件推导整体存在性与爆破结果，并引入新测试函数获得了局部解寿命的精确上下界估计。
This paper is concerned with the blowup phenomena for initial-boundary value problem for certain semi linear parabolic, dispersive and hyperbolic equations in cone-like domain. The result proposes a unified treatment of estimates for lifespan of solutions to the problem by test function method. The Fujita exponent p=1 + 2/N appears as a threshold of blowup phenomena for small data when $C_{Σ}=R^N$ , but the case of cone-like domain with boundary the threshold changes and explicitly given via the first eigenvalue of corresponding Laplace-Beltrami operator with Dirichlet boundary condition as in…
We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-σ}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range of parameters $A,B,σ,p\in\R$ and $C>0$. We provide a complete characterization of the set of $(p,σ)\in\R^2$ such that the equation has no positive (super-)solutions, depending on the values of $A,B$ and the principle Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the ana…
Fujita-type blow-up for inhomogeneous semilinear heat equations with regularly varying forcing
利用正则变化理论，通过空间质量函数的变差指数刻画临界指数，建立了非齐次半线性热方程Fujita型爆破的统一框架，并证明了次临界范围全局解不存在及临界情况爆破。
A fast reconstruction algorithm for geometric inverse problems using topological sensitivity analysis and Dirichlet-Neumann cost functional approach
本文提出了一种基于Kohn-Vogelius公式和拓扑灵敏度分析的快速算法，用于从边界测量中重建浸入各向异性介质中的物体，通过拓扑优化方法最小化能量型函数，利用拓扑渐近展开重构未知物体。
Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces
本文研究了双曲空间上分数热方程的Fujita指数及半线性分数椭圆方程的解的存在性，通过新的分数Poincaré不等式和相关紧嵌入定理得出关键结果。
Well-posedness and regularity theory for the fractional diffusion-wave equation in Lebesgue spaces
本文研究半线性分数阶扩散-波动方程，在拉普拉斯算子相关分数幂尺度下给出线性算子族估计，分析局部mild解的存在唯一性与最大区间延拓性，还探讨空间正则性及解对初值的连续依赖性。
Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in $R^n$
证明了全空间中超临界半线性热方程整体经典解的有界性与最佳时间衰减性，基于Giga-Kohn型加权能量估计与临界Morrey空间分析，揭示了Morrey空间在整体解结构中的关键作用。
Solitary waves for the power degenerate NLS -- existence and stability
本文研究幂退化二阶微分算子驱动的半线性薛定谔方程，利用 Caffarelli-Kohn-Nirenberg 不等式在精确参数范围内构造孤立波，证明其正性、光滑性与几乎指数衰减，并完整刻画其谱稳定性，同时提出基态径向性、非退化性与唯一性等未解猜想。
Well-posedness and Blow-up in a semilinear heat equation with variable-order Scarpi memory
研究变阶Scarpi记忆半线性热方程的适定性与爆破，通过Kaplan加权矩方法建立有限时间爆破准则，并给出寿命估计与数值验证。
We consider the Klein--Gordon equation associated with the Laplace--Beltrami operator $Δ$ on real hyperbolic spaces of dimension $n\!\ge\!2$; as $Δ$ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well--posedness results for the corresponding semilinear equation with low regularity data.
Concavity and perturbed concavity for $p$-Laplace equations
本文研究 $p$-Laplace 方程解的凹性，在次齐次、奇异及超齐次情形下给出继承或近似凹性结果，并推广至分数阶情形，部分结果在半线性情形下也是新的。
Quasilinear Schrödinger Critical Problem on the Heisenberg group \(\mathbb{H}^N\)
研究海森堡群上特定拟线性薛定谔方程驻波解的存在性，通过非线性变量变换转化方程，运用变分方法及集中紧致性论证，在福兰德 - 斯坦索伯列夫空间中建立了非平凡解的存在性。
Global Future Existence for a Nonlinear Wave Equation in a (1+3+n)-dimensional Cosmological Supersymmetric Spacetime
该研究针对1+n维宇宙学超对称时空中的半线性波动方程柯西问题，基于新型修正能量证明小初值解的全局未来存在性，其框架可推广至含爱因斯坦方程的张量波动方程。
Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality
本文研究具单调系数的临界p-拉普拉斯方程，完成其正整体解的刘维尔分类，建立Schoen型哈纳克不等式，为经典半线性方法提供拟线性对应及替代证明。
Concentration phenomena for a mixed local/nonlocal Schrödinger equation with Dirichlet datum
本文研究含局部 Laplace 项与非局部分数阶 Laplace 项的小参数 Schrödinger 方程，构造在内部点集中的解，并揭示非局部效应使约化能量呈多项式衰减。
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