Martino Prizzi
We prove the existence of a compact L^2-H^1 attractor for a reaction-diffusion equation in R^n. This improves a previous result of B. Wang concerning the existence of a compact L^2-L^2 attractor for the same equation.
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
We consider a reaction-diffusion system which may serve as a model for a ferment catalytic reaction in chemistry. The model consists of a system of reaction diffusion equations with unbounded time dependent coefficients and different polynomial reaction terms. An exponential decay of the globally bounded solutions is proved. The key tool of the proofs are properties of analytic semigroups and some inequalities.
In this note we propose the exact artificial boundary conditions formula to the fractional reaction-subdiffusion equation on an unbounded domain. With the application of Laplace transformation, the exact artificial boundary conditions (ABCs) are derived to reformulate the original problem on the unbounded domain to an initial-boundary-value problem on the bounded computational domain. By the Kreiss theory, we prove that the reduced initial-boundary value problem is stability. Based on the properties of tempered fractional calculus, we obtain that the reduced initial-boundary value problem is l…
We present a unified approach to characterising fast-reaction limits of systems of either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit $k \to \infty$ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line.…
In this paper we consider second order evolution equations with unbounded dynamic feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.
Exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain
本文研究无界域上非局部时滞反应扩散方程的指数吸引子，通过结合系统的挤压性质与 Banach 空间覆盖引理，直接在自然相空间中构造出具有明确分形维数的指数吸引子，推广了已有方法。
We prove existence of global attractors for parabolic equations of the form $$u_t+β(x)u-\sum_{ij}\partial_i(a_{ij}(x)\partial_j u)=f(x,u)$$ with Dirichlet boundary condition on an arbitrary unbounded domain $Ω$ in $\R^3$, without smoothness assumptions on $a_{ij}(\cdot)$ and $\partialΩ$.
In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type $du = (Au + f(u))dt + σ(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $σ$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function $f$ possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence …
We consider a nonlinear reaction-diffusion equation settled on the whole euclidean space. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L^2. Then we adapt the short trajectory method to establish the existence of the global attractor and, if the space dimension is at most 3, we also find an upper bound of its Kolmogorov's entropy.
Minimal speed of unbounded traveling wave solutions for a 1D reaction-diffusion equation and their relationship with the dynamics at infinity
本文研究了一维反应扩散方程中无界行波解的最小速度及其与无穷远处动力学的关系，通过Poincaré型紧致化方法揭示了二维常微分方程组的动力学特性，确定了行波解的分类、正负性及最小速度的显式形式。
On boundary regularity for the fractional p-Laplacian with unbounded reactions
该研究针对带无界反应项的分数阶p-拉普拉斯椭圆方程，证明了其解的边界赫尔德正则性，结果几乎最优并推广了线性情形的相关定理。
Spreading speeds and spreading sets of reaction-diffusion equations
本文针对一般无界初始支集的反应扩散方程，建立任意方向传播速度的变分公式并刻画水平集渐近形状，同时给出传播条件与正速度行波前沿存在性的联系。
Invariant measures, periodic measures and pullback measure attractors of McKean-Vlasov stochastic reaction-diffusion equations on unbounded domains
本文研究了R^n上非自治McKean-Vlasov随机反应扩散方程的长期动力学，证明了拉回测度吸引子、不变测度与周期测度的存在唯一性，并建立了系统收敛时的上半连续性与测度收敛性。
Well-posedness and large deviations of fractional McKean-Vlasov stochastic reaction-diffusion equations on unbounded domains
本文研究无界区域上带任意次多项式漂移的分数阶McKean-Vlasov随机反应扩散方程，证明了其适定性与受控方程解的强收敛性，最终通过弱收敛方法在弱假设下建立了大偏差原理。
The primary goal of this paper is to characterize solutions to coupled reaction-diffusion systems. Indeed, we use operators theory to show that under suitable assumptions, then the solutions to the reaction-diffusion equations exist. As applications, we consider a mathematical model arising in Biology and in Chemistry.
We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball $$ u_t=Δu^m+a(x)u^p, $$ for $m>0$, $0<p\le\max\{1,m\}$, $a(x)=\mathds{1}_{B_L}(x)$, $0<L<\infty$ and $N\ge2$. We study when solutions, which are global in time, are bounded or unbounded. In particular we show that the precise value of the length $L$ plays a crucial role in the critical case $p=m$ for $N\ge3$. We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when $L=\infty$.
We study the long time behavior, as $t\to\infty$, of solutions of $$ \left\{ \begin{array}{ll} u_t = u_{xx} + f(u), & x>0, \ t >0,\\ u(0,t) = b u_x(0,t), & t>0,\\ u(x,0) = u_0 (x)\geqslant 0 , & x\geqslant 0, \end{array} \right. $$ where $b\geqslant 0$ and $f$ is an unbalanced bistable nonlinearity. By investigating families of initial data of the type $\{ σϕ\}_{σ>0}$, where $ϕ$ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value $σ^*$ such that the solution conv…
We consider Leray's problem on stationary Navier-Stokes flows with arbitrary large fluxes in an unbounded cylinder with several exits to infinity. For a stationary Navier-Stokes flow with large fluxes in the unbounded cylinder we prove that, if the difference between the pressure of the main flow and the pressure of the Poiseuille flow with the same flux in a branch of the cylinder satisfies some asymptotic boundedness condition at $|x|\rightarrow \infty$ then the flow behaves at infinity of the branch like the Poiseuille flow.
Reaction-diffusion equations in periodic media: spreading speeds and spreading sets
本文研究周期介质中反应-扩散-平流方程的解，证明了传播集合的存在性及传播速度的变分公式，并利用回缩解估计无界初始支撑解的上水平集位置，适用于Fisher-KPP、点火和双稳态反应。
The existence of two nontrivial smooth solutions to a semilinear Robin problem with indefinite unbounded potential and asymmetric nonlinearity $f$ is established. Both crossing and resonance are allowed. A third nonzero solution exists provided $f$ is $C^1$. Proofs exploit variational methods, truncation techniques, and Morse theory.
Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain
研究无界域上延迟反应扩散方程全局吸引子的存在性及Hausdorff和分形维数上界，通过先验估计和渐近紧性证明存在性，并利用解分解与挤压性质获得仅依赖方程内在特征的维数估计。
In this paper, we present an approach to characterising fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form uxx in usual reaction-diffusion equation, which represent linear diffusion, by terms of form phi(u)xx, representing nonlinear diffusion. We prove the convergence as k tends to infinity to a limit that is determined by the unique solution of a certain scalar nonlinear diffusion limit problem.
The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of $L^p$-viscosity solutions is derived from the weak Harnack inequality for $L^p$-viscosity supersolutions. The local maximum principle for $L^p$-viscosity subsolutions and the Harnack inequality for $L^p$-viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.
The paper is devoted to the investigation of a reaction-diffusion system of equations describing the process of blood coagulation. Existence of pulses solutions, that is, positive stationary solutions with zero limit at infinity is studied. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray-Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.
This paper is concerned with diffusion-reaction equations where the classical diffusion term, such as the Laplacian operator, is replaced with a singular integral term, such as the fractional Laplacian operator. As far as the reaction term is concerned, we consider bistable non-linearities. After properly rescaling (in time and space) these integro-differential evolution equations, we show that the limits of their solutions as the scaling parameter goes to zero exhibit interfaces moving by anisotropic mean curvature. The singularity and the unbounded support of the potential at stake are both …
We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincaré's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.
Let $Ω$ be a domain with noncompact boundary. It is known that the Helmholtz decomposition is not always valid in $L^p(Ω)$ except for the energy space $L^2 (Ω)$. In this paper we consider a typical unbounded domain whose boundary is given as a Lipschitz graph, and show that the Helmholtz decomposition holds in certain anisotropic spaces which include some infinite energy vector fields.
This paper is concerned with the existence and further properties of propagation speeds of transition fronts for bistable reaction-diffusion equations in exterior domains and in some domains with multiple cylindrical branches. In exterior domains we show that all transition fronts with complete propagation propagate with the same global mean speed, which turns out to be equal to the uniquely defined planar speed. In domains with multiple cylindrical branches, we show that the solutions emanating from some branches and propagating completely are transition fronts propagating with the unique pla…
A semilinear singularly perturbed reaction-diffusion equation with Dirichlet boundary conditions is considered in a convex unbounded sector. The singular perturbation parameter is arbitrarily small, and the "reduced equation" may have multiple solutions. A formal asymptotic expansion for a possible solution is constructed that involves boundary and corner layer functions. For this asymptotic expansion, we establish certain inequalities that are used in a subsequent paper to construct sharp sub- and super-solutions and then establish the existence of a solution to a similar nonlinear elliptic p…
We construct nontrivial entire solutions for a bistable reaction-diffusion equation in a class of domains that are unbounded in one direction. The motivation comes from recent results of Berestycki, Bouhours, and Chapuisat concerning propagation and blocking phenomena in infinite domains. A key assumption in their study was the "cylinder-like" assumption: their domains are supposed to be straight cylinders in a half space. The purpose of this paper is to consider domains that tend to a straight cylinder in one direction. We also prove the existence of an entire solution for a one-dimensional p…
Coupling a reaction-diffusion equation with ordinary differential equations (ODE) may lead to diffusion-driven instability (DDI) which, in contrast to the classical reaction-diffusion models, causes destabilization of both, constant solutions and Turing patterns. Using a shadow-type limit of a reaction-diffusion-ODE model, we show that in such cases the instability driven by nonlocal terms (a counterpart of DDI) may lead to formation of unbounded spike patterns.
Freidlin-Gärtner formula and asymptotic profile in reaction-diffusion equations
研究反应扩散方程在周期性介质中大时间行为，推导弗里德林-加尔特纳公式并探讨弱双稳方程的渐进行为，包括正则公式和脉动行波收敛性。
Unbounded solutions for the Muskat problem
本文证明了稳定机制下Muskat问题具有在无穷远处二次增长的局部解，通过能量方法在Sobolev空间获得界，并克服了无穷远处Rayleigh-Taylor条件缺失及核新结构等挑战。
We classify the self-similar blow-up profiles for the following reaction-diffusion equation with critical strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $0<p<1$ such that $m+p=2$ and $σ>2$ completing the analysis performed in a recent work where this very interesting critical case was left aside. We show that finite time blow-up solutions in self-similar form exist for $σ>2$. Moreover all the blow-up profiles have compact support and their supports are \emph{localized}: there exists an explicit $η>0$…
The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue …
Convergence of solutions for a reaction-diffusion problem with fractional Laplacian
本文研究无界区域上带分数阶Laplacian的非局部反应扩散方程，证明分数阶参数趋于1时其解收敛于经典标准Laplacian方程的解，还探究了全局吸引子存在性，为该类问题的解收敛性研究提供了首个相关结果。
Spreading, flattening and logarithmic lag for reaction-diffusion equations in R^N: old and new results
本文研究R^N中反应扩散方程解的大时间行为，综述传播速度与渐近对称性，推导水平集平坦化性质，并针对对数增长的无界初始支集证明新的对数滞后估计，表明其滞后与紧支撑情形相同。
We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in any space dimension $x\in\mathbf{R}^N$, $t\geq0$ and with exponents $m>1$, $p\in(0,1)$ and $σ>2(1-p)/(m-1)$. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are \emph{com…
We consider a nonlinear integro-differential equation for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The equation can be written as a quasilinear Cauchy problem. For bounded reaction rates we prove global existence and uniqueness of classical solutions by means of evolution operator theory. We also prove global existence of weak solutions for unbounded reaction rates by a compactness argument.
We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between "no acceleration and acceleration". This implies in particular that, for tails exponentially unbounded but lighter than algebraic , acceleration never occurs in presence of an Allee effect. This is in…
We consider reaction-diffusion equations $\partial_tu=Δu+f(u)$ in the whole space $\mathbb{R}^N$ and we are interested in the large-time dynamics of solutions ranging in the interval $[0,1]$, with general unbounded initial support. Under the hypothesis of the existence of a traveling front connecting $0$ and $1$ with a positive speed, we discuss the existence of spreading speeds and spreading sets, which describe the large-time global shape of the level sets of the solutions. The spreading speed in any direction is expressed as a Freidlin-Gärtner type formula. This formula holds under general …
We discuss conditions for well-posedness of the scalar reaction-diffusion equation $u_{t}=Δu+f(u)$ equipped with Dirichlet boundary conditions where the initial data is unbounded. Standard growth conditions are juxtaposed with the no-blow-up condition $\int_{1}^{\infty}1/f(s) \d s=\infty$ that guarantees global solutions for the related ODE $\dot u=f(u)$. We investigate well-posedness of the toy PDE $u_{t}=f(u)$ in $L^{p}$ under this no-blow-up condition. An example is given of a source term $f$ and an initial condition $ψ\in L^{2}(0,1)$ such that $\int_{1}^{\infty}1/f(s)\d s=\infty$ and the t…
We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)$ being the characteristic function of an interval. we prove that there exists $p_0=\max\{1,\frac{m+1}2\}$ such that all global solution are bounded if $p>p_0$, while for $p\le p_0$ all the solution are global and unbounded. In the last case, we prove that if $p<m$ the grow-up rate is different to the one obtained when $a(x)\equiv1$, while if $p>m$ the grow-up rate coincides with that rate, but only inside the support of $a$; outside the interva…
In this paper, we study the existence and stability of travelling wave solutions of a kinetic reaction-transport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves. The minimal speed of propagation of waves is obtained from an explicit dispersion relation. We construct the waves using a technique of sub- and supersolutions and prove their \eb{weak} stability in a we…
We study the long time behavior of solutions of the non-autonomous Reaction-Diffusion equation defined on the entire space R^n when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L^2(R^n) and H^1(R^n), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.
This paper is concerned with the existence and uniqueness of transition fronts of a general reaction-diffusion-advection equation in domains with multiple branches. In this paper, every branch in the domain is not necessary to be straight and we use the notions of almost-planar fronts to generalize the standard planar fronts. Under some assumptions of existence and uniqueness of almost-planar fronts with positive propagating speeds in extended branches, we prove the existence of entire solutions emanating from some almost-planar fronts in some branches. Then, we get that these entire solutions…
Reaction-diffusion equations in periodic media: convergence to pulsating fronts
研究周期性介质中反应扩散-对流方程的长期行为，证明在弱稳定性和脉动前沿存在性假设下，解的长期动力学中前沿轮廓出现，无论初始支持有界还是无界，并探讨了入侵解的渐近入侵形状和几何性质。
Acceleration or finite speed propagation in weakly monostable reaction-diffusion equations
本文研究弱单稳态反应扩散方程的传播现象，分析初始数据衰减率对传播速率的影响，推导有限速度传播与加速的分界，给出水平集位置估计和精确加速指数，并通过数值模拟验证结果。
In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than $(T-t)^{-1}$, the expected one. Moreover, we show that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.
The existence of a global attractor for wave equations in unbounded domains is a challenging problem due to the non-compactness of the Sobolev embeddings. To overcome this difficulty, some authors have worked with weighted Sobolev spaces which restrict the choice of the initial data. Using the "tail estimation method" introduced by B. Wang for reaction diffusion equations, we establish in this paper the existence of a global attractor for two wave equations in the traditional Hilbert spaces $\displaystyle H^1(Ω)\times L^2(Ω)$ where $Ω$ is an unbounded domain of $\R^N$. The first equation, with…
Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction
该研究针对带加权反应的拟线性反应扩散方程，分类了自相似形式的爆破解，揭示了非齐次权重对爆破模式的影响，丰富了反应扩散方程爆破行为的研究。
Large Deviations of Invariant Measures of Fractional Reaction-Diffusion Equations Driven by Multiplicative Noise on R^n
本文针对R^n上带多项式反应项、由无界乘性噪声驱动的分数阶随机反应扩散方程，建立其不变测度的大偏差，为该类无界域方程不变测度大偏差的首个相关结果。
Some remarks on spatial uniformity of solutions of reaction-diffusion PDE's and a related synchronization problem for ODE's
本文研究了反应扩散PDE解的空间均匀性条件，基于雅可比矩阵和拉普拉斯算子的第一狄利克雷特征值，同时推导了ODE网络同步问题的类比结果。
We study a reaction-diffusion-convection problem with nonlinear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this nonlinearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model e…
The asymptotic behavior of solutions of two dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations in unbounded domains is discussed in this work (for example, Poincaré domains). We first prove the existence of $\mathbb{H}^1$-random attractors for the stochastic flow generated by 2D SCBF equations (for the absorption exponent $r\in[1,3]$) perturbed by an additive noise on Poincaré domains. Furthermore, we deduce the existence of a unique invariant measure in $\mathbb{H}^1$ for the 2D SCBF equations defined on Poincaré domains. In addition, a remark on the extension of these…
Remarks on the linear wave equation
本文对线性波动方程进行了多方面探讨，包括弱解存在唯一性、能量方程满足、解的增长性质、有界到无界域的过渡及解表示的调和。
Unbounded Sturm attractors for quasilinear equations
该研究针对解可能无限时间爆破的拟线性抛物方程，分析其无界全局吸引子的渐近动力学，揭示无穷远梯度动力学结构、扩散与反应的维度竞争作用，证明双曲平衡点间异宿轨的存在，且这类吸引子可描述特定黑洞事件视界的容许初值空间。
We study the self-similar blow-up profiles associated to the following second order reaction-diffusion equation with strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $0<p<1$ and $σ>2(1-p)/(m-1)$. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for $m+p>2$ and $σ$ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when $p<1$. We moreover prove that, if the con…
Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=Δu^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in the range of exponents $1\leq p<m$ and dimension $N\geq3$. The self-similar solution is unbounded at $x=0$ and has a logarithmic vertical asymptote, but it remains bounded at any $x\neq0$ and $t\in(0,\infty)$ and it is a weak solution in $L^1$ sense, which moreover satisfies $u(t)\in L^p(\real^N)$ for any $t>0$ and $p\in[1,\infty)$. As an application of this self-si…
Large Deviations of Fractional Stochastic Equations with Non-Lipschitz Drift and Multiplicative Noise on Unbounded Domains
本文研究无界域上具任意阶多项式漂移和乘性噪声的非局部分数阶随机反应扩散方程的大偏差原理，通过弱收敛方法并结合一致尾部估计克服Sobolev嵌入非紧性难题，最终建立该方程的大偏差。
This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = Δv(t,x) - v(t,x) + \int_{\R^d}K(y)g(v(t-h,x-y))dy, x \in \R^d,\ t >0;$ where $h>0$ and $d\in\Z_+$. We give two general results for $d\geq1$: on the global stability of semi-wavefronts in $L^p$-spaces with unbounded weights and the local stability of planar wavefronts in $L^p$-spaces with bounded weights. We also give a global stability result for $d=1$ which includes the global stability on Sobolev spaces. Here $g$ is not assumed to be monotone and the kernel $K$ is not …
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。