Uniqueness of sign-changing solutions to Trudinger's equation
Riku Anttila, Peter Lindqvist, Mikko Parviainen
研究Trudinger抛物方程在时间依赖C²边界数据下的sign-changing解唯一性问题，采用分析方法证明解的唯一性。
本文建立了Trudinger抛物方程在时间依赖$C^2$狄利克雷边界数据下的sign-changing解唯一性.
We establish uniqueness for sign-changing solutions to Trudinger's parabolic equation with time dependent $C^2$ Dirichlet boundary data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Given a sufficiently symmetric domain $Ω\Subset\mathbb{R}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $β>4πk$ we construct blowing-up solutions $(u_\varepsilon)\subset H^1_0(Ω)$ to the Moser-Trudinger equation such that as $\varepsilon\downarrow 0$, we have $\|\nabla u_\varepsilon\|_{L^2}^2\to β$, $u_\varepsilon \rightharpoonup u_0$ in $H^1_0$ where $u_0$ is a sign-changing solution of the Moser-Trudinger equation and $u_\varepsilon$ develops $k$ positive spherical bubbles, all concentrating at $0\in Ω$. These $3$ features (lack of quantization, non-zero weak limit and bubble clustering) …
Constrained Moser-Trudinger-Onofri inequality and a uniqueness criterion for the mean field equation
本文建立了二阶矩偏离0约束下的Moser-Trudinger-Onofri不等式，并证明该偏离阈值是常数$a$接近$\frac{1}{2}$时平均场方程的唯一性准则。
Trudinger's Parabolic Equation
研究非负解的唯一性，采用Galerkin方法获得基本估计。
In this work we study the existence of nodal solutions for the problem $$ -Δu = λu e^{u^2+|u|^p} \text{ in }Ω, \; u = 0 \text{ on }\partial Ω, $$ where $Ω\subseteq \mathbb R^2$ is a bounded smooth domain and $p\to 1^+$. If $Ω$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^+$, when $Ω$ is an arbitrary domain and $λ$ is small enough. As far as we know, this is the first construction of sign-c…
Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\widetilde u$ the problem is well-posed, b) for initial data above this threshold function $…
We are concerned with the Moser-Trudinger problem \begin{equation*} \begin{cases} -Δu=λue^{u^2}~~&\mbox{in}~Ω,\\[0.5mm] u>0 ~~ &{\text{in}~Ω},\\[0.5mm] u=0~~&\mbox{on}~\partial Ω, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^2$ is a smooth bounded domain and $λ>0$ is sufficiently small. Qualitative analysis for Moser-Trudinger nonlinearities has been studied in recent decades, however there is still a lot of clarity about this issue, even for a low energy. The reason is that this problem is a critical exponent for dimension two and will lose compactness. Here by using a variety of lo…
We construct a singular solution of a stationary nonlinear Schrödinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.
Local Hölder regularity for bounded, signed solutions to nonlocal Trudinger equations
本文针对非局部 Trudinger 方程，在 $1<p<\infty$、$s\in(0,1)$ 范围内证明了有界变号弱解的局部 Hölder 正则性，克服了 Tail 项导致变量替换不可用的困难，采用了近年来发展的新方法。
We investigate the next Trudinger-Moser critical equations, \[ \begin{cases} -Δu=λue^{u^2+α|u|^β}&\text{ in }B,\\ u=0&\text{ on }\partial B, \end{cases} \] where $α>0$, $(λ,β)\in(0,\infty)\times(0,2)$ and $B\subset \mathbb{R}^2$ is the unit ball centered at the origin. We classify the asymptotic behavior of energy bounded sequences of radial solutions. Via the blow--up analysis and a scaling technique, we deduce the limit profile, energy, and several asymptotic formulas of concentrating solutions together with precise information of the weak limit. In particular, we obtain a new necessary cond…
In this paper, we will solve this uniqueness problem of positive solutions to the following equations of exponential growth: \begin{equation*} \begin{cases} -Δu =λue^{u^2},\quad\quad & x\in B_1\subset \mathbb{R}^2,\\ u>0, & x\in B_1,\ \\ u=0,\quad\quad &x\in \partial B_1, \end{cases} \end{equation*} where $ 0<λ<λ_1(B_1)$ and $λ_1(B_1)$ denotes the first eigenvalue of the operator $-Δ$ with the Dirichlet boundary in unit disk. Our method relies on delicate and difficult analysis of radial solutions to the above equation and careful asymptotic expansion of solutions near the boundary. This uniqu…
Existence of sign-changing solutions for a logarithmic weighted Kirchhoff problem in the whole of $\mathbb{R}^{N}$ with exponential growth non-linearity
本文研究全空间中具有指数增长非线性的对数加权 Kirchhoff 问题，利用 Nehari 流形约束极小化、定量形变引理和度理论证明了低能量变号解的存在性。
We establish the uniqueness of the higher radial bound state solutions of $$ Δu +f(u)=0,\quad x\in \RR^n. \leqno(P) $$ We assume that the nonlinearity $f\in C(-\infty,\infty)$ is an odd function satisfying some convexity and growth conditions, and either has one zero at $b>0$, is non positive and not identically 0 in $(0,b)$, and is differentiable and positive $[b,\infty)$, or is positive and differentiable in $[0,\infty)$.
We study sign-changing radial solutions for the following semi-linear elliptic equation \begin{align*} Δu-u+|u|^{p-1}u=0\quad{\rm{in}}\ \mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N), \end{align*} where $1<p<\frac{N+2}{N-2}$, $N\geq3$. It is well-known that this equation has a unique positive radial solution and sign-changing radial solutions with exactly $k$ nodes. In this paper, we show that such sign-changing radial solution is also unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions are non-degenerate, i.e., the kernel of the linearized operator is exactly $N$-dimensional.
Let $n \geq 2$ and $Ω\subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(Ω)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear $n$-Laplace equation with critical or sub-critical exponential nonlinearity in a ball $B(R)$ with dirichlet boundary condition. In this paper, we prove that under suitable growth conditions on the nonlinearity, there exists an $γ_0 > 0$, and a corresponding $R_0(γ_0 ) > 0$ such that for all $0 < R < R_0$ , the problem admits a unique non degenerate positive radial solution $u$…
Quantization analysis of Moser-Trudinger equations in the Poincaré disk and applications
本文研究了庞加莱盘上Moser-Trudinger方程的正解性质，证明了在特定条件下存在正临界点，并在λ接近0时证明方程仅有一个正解。
Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators
研究混合局部与非局部算子的Choquard方程，通过Nehari流形方法证明最小能量正解的存在性，并利用下降流不变集建立变号解的存在性，当非线性项为奇函数时存在无穷多变号解。
We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profil…
We prove that two Aleksandrov solutions of a generated prescribed Jacobian equation have the same gradients at points where they are both differentiable. For the optimal transportation case where two solutions can be translated to agree at a point without changing the $g$-subdifferential at that point, we recover the uniqueness up to a constant of solutions. For the general case, our result is a new proof with less regularity assumptions of a key theorem recently used to prove the uniqueness of solutions.
A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions
本文研究三维单位球上耦合薛定谔系统的变号径向解，通过建立新的刘维尔型定理与有界性结果，证明了参数满足一定条件时具有指定变号次数的径向解的唯一性。
An existence and uniqueness result to evolution equations with sign-changing pseudo-differential operators and its applications to logarithmic Laplacian operators and second-order differential operators without ellipticity
本文通过扩展 Fourier 变换定义域构建新弱形式，解决了带变号伪微分算子的发展方程的存在与唯一性问题，并应用于对数 Laplace 算子和无椭圆性的二阶微分算子。
In the present article we investigate a boundary problem with non-local conditions for mixed parabolic-hyperbolic type equation with three lines of type changing. Considered mixed domain contains a rectangle as a parabolic part and three domains bounded by smooth curves and by type-changing lines as a hyperbolic part of the mixed domain. We prove the uniqueness applying energy integral method. The proof of the existence will be done by reducing the original problem into the system of the second kind Volterra integral equations.
Uniqueness and non-uniqueness of least energy normalized solutions for nonlinear Schrödinger equations on compact metric graphs
该研究探讨紧度量图上非线性薛定谔方程的最小能量归一化解的唯一性，通过分析不同幂次与图类，揭示其唯一性依赖于非线性幂次、图拓扑及度量，且常号与变号解存在结构差异。
We study solutions of the generalized porous medium equation on infinite graphs. For nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild solutions on any graph. For changing sign integrable data, we show existence and uniqueness under extra assumptions such as local finiteness or a uniform lower bound on the node measure.
We show weak-strong uniqueness and stability results for the motion of a two or three dimensional fluid governed by the Navier-Stokes equation interacting with a flexible, elastic plate of Koiter type. The plate is situated at the top of the fluid and as such determines the variable part of a time changing domain (that is hence a part of the solution) containing the fluid. The uniqueness result is a consequence of a stability estimate where the difference of two solutions is estimated by the distance of the initial values and outer forces. For that we introduce a methodology that overcomes the…
Let $1<q<p$ and $a\in C(\overlineΩ)$ be sign-changing, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$. We show that the functional \[ I_{q}(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] has exactly one nonnegative minimizer $U_{q}$ (in $W_{0}^{1,p}(Ω)$ or $W^{1,p}(Ω)$). In addition, we prove that $U_{q}$ is the only possible \textit{positive} solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if $q$ is close enough to $p$ then $U_{q}$ is positive, which als…
Existence results for Toda systems with sign-changing prescribed functions: Part II
本文研究紧黎曼曲面上带变号预设函数的临界Toda系统，通过改进Moser-Trudinger不等式、结合Brezis-Merle型分析与Pohozaev恒等式，证明Jost-Lin-Wang的解存在充分条件在$ρ_1=ρ_2=4π$且函数变号时仍成立。
We study a singular nonlinear ordinary differential equation on intervals $[0,R)$ with $R\le +\infty$, motivated by the Ginzburg-Landau models in superconductivity and Landau-de Gennes models in liquid crystals. We prove existence and uniqueness of positive solutions under general assumptions on the nonlinearity. Further uniqueness results for sign-changing solutions are obtained for a physically relevant class of nonlinearities. Moreover, we prove a number of fine qualitative properties of the solution that are important for the study of energetic stability.
Nodal solutions for Logarithmic weighted $(N, p)$-Laplacian problem with exponential nonlinearities
本文研究对数加权 $(N,p)$-Laplacian 问题在单位球上的最小能量变号解的存在性，通过 Nehari 集约束极小化、定量形变引理和度理论，在指数非线性条件下证明了存在性。
A Fractional Logistic-Type Elliptic Problem with Global Interactions: Existence, Local Uniqueness, and the Fractional-to-Local Limit
本文研究带全局相互作用的分数阶逻辑型椭圆问题，通过极小化与不动点定理证明解的存在性，并在特定条件下建立局部唯一性及分数阶到局部极限的收敛性。
We study a nonlocal version of the two-phase Stefan problem, which models a phase transition problem between two distinct phases evolving to distinct heat equations. Mathematically speaking, this consists in deriving a theory for sign-changing solutions of the equation, ut = J * v - v, v = Γ(u), where the monotone graph is given by Γ(s) = sign(s)(|s|-1)+ . We give general results of existence, uniqueness and comparison, in the spirit of [2]. Then we focus on the study of the asymptotic behaviour for sign-changing solutions, which present challenging difficulties due to the non-monotone evoluti…
Limiting Absorption Principle for the Helmholtz Equation with Sign-Changing Coefficients in Multilayer Spheres
针对多层球壳中折射率正负交替的亥姆霍兹问题，通过构造T-强制算子恢复强制性，并集成复波数Dirichlet-to-Neumann算子处理无界域紧性缺失，结合变分框架与先验估计严格建立极限吸收原理，证明传输问题的适定性，并通过最优迹常数分析唯一性对几何的依赖，为多层超材料设计提供数学准则。
In this paper diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudo-differential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diffusion modes change. These memory effects are classified and studied in detail. Processes that have distinctive regimes of different types of diffusion depending on time are ubiquitous i…
We study a simplified system of the original Ericksen--Leslie equations for the flow of nematic liquid crystals. This is a coupled non-parabolic dissipative dynamic system. We show the convergence of global classical solutions to single steady states as time goes to infinity (uniqueness of asymptotic limit) by using the Łojasiewicz--Simon approach. Moreover, we provide an estimate on the convergence rate. Finally, we discuss some possible extensions of the results to certain generalized problems with changing density or free-slip boundary condition.
We study the existence and nonexistence of a Cauchy problem of the semilinear heat equation $\partial_tu=Δu+|u|^{p-1}u$ in $\mathbb{R}^N\times(0,T)$, $u(x,0)=ϕ(x)$ in $\mathbb{R}^N$, in $L^1(\mathbb{R}^N)$. Here, $N \ge 1$, $p=1+2/N$ and $ϕ\in L^1( \mathbb{R}^N)$ is a possibly sign-changing initial function. Since $N(p-1)/2=1$, the $L^1$ space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every $ϕ\in L^1(\mathbb{R}^N)$. Let $X_q:=\{ ϕ\in L^1_{\rm{loc}}(\mathbb{R}^N)\ |\ \int_{\mathbb{R}^N}|ϕ| \left[\log (e+|ϕ|)…
Inverse problem for the subdiffusion equation with non-local in time condition
本文研究带Caputo分数阶导数和时间非局部条件的次扩散方程反问题，在右端为fg(t)形式下证明g(t)不变号时反问题解存在唯一，并讨论变号情形下的唯一性破坏、适定性条件及正交性条件，结果对经典扩散方程亦为新的。
We provide existence and uniqueness of renomalized solutions to a general nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in $\mathbb{R}^n$. Namely we study \begin{equation*} \left\{\begin{array}{l } \partial_t u-{\rm div} A(t,x,\nabla u)= f(t,x) \in L^1(Ω_T),\\ u(0,x)=u_0(x)\in L^1(Ω). \end{array}\right. \end{equation*} The growth of the monotone vector field $A$ is assumed to be controlled by a generalized nonhomogeneous and anisotropic $N$-function $M:[0,T)\times Ω\times\mathbb{R}^n \to[0,\infty)$. Existence and uniqueness of renormalized solutions are…
Parabolic Frequency for Doubly Nonlinear Equations on Manifolds
本文研究了流形上双重非线性抛物方程的抛物线频率单调性公式，得出了关于退化唯一性和无限远处的唯一延续性结果，并建立了古典型解的利奥维勒型定理。
We present a modified version of the two-player "tug-of-war" game introduced by Peres, Schramm, Sheffield, and Wilson. This new tug-of-war game is identical to the original except near the boundary of the domain $\partial Ω$, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results. We show that the finite difference equation has unique maximal and minim…
Stability and Reconstruction of a Nonlinearity in a Parabolic Equation from Partial Boundary Data
研究从边界任意子集的单次通量确定非线性抛物型方程半线性项的反问题，建立赫尔德型稳定性估计，提出迭代重构算法并通过数值实验验证其准确性。
A three-block higher-rank deformed Hermitian--Yang--Mills correspondence on $\mathbb P^1\times\mathbb P^1$
本研究针对$\mathbb P^1\times\mathbb P^1$上的链$\mathcal{O}(1)\to\mathcal{O}(2)^{\oplus2}\to\mathcal{O}(3)$，探究带垂直权重的四阶不变高阶变形Hermitian–Yang–Mills方程，通过分析解的存在条件与稳定性，完成了相关解的分类并给出了端点的光滑性证明。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。