Liviu I. Ignat, Ademir Pazoto
In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions.
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
Long-time asymptotics for nonlocal porous medium equation with absorption or convection
本文针对带吸收或对流项的非局部多孔介质方程，在非局部扩散占优的参数区域内，通过调整熵方法推导出相似变量下解的相对熵呈指数收敛，明确了该类方程的长时间渐近行为。
In this paper we consider a nonlocal viscous Burgers equation and study the well-posedness and asymptotic behaviour of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.
We consider the "convection-diffussion" equation $u_t=J*u-u-uu_x,$ where $J$ is a probability density. We supplement this equation with step-like initial conditions and prove a convergence of corresponding solution towards a rarefaction wave, i.e. a unique entropy solution of the Riemann problem for the nonviscous Burgers equation. Methods and tools used in this paper are inspired by those used in [Karch, Miao and Xu, SIAM J. Math. Anal. {\bf 39} (2008), no. 5, 1536--1549.], where the fractal Burgers equation was studied.
Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection
研究带对流项的非局部热方程非负解的大时间行为，发现其大时间极限由对流与扩散相对强度决定的极限方程的源型解给出，并确定了各向异性扩散算子在正交子空间上的投影。
In this paper we consider a subcritical model that involves nonlocal diffusion and a classical convective term. In spite of the nonlocal diffusion, we obtain an Oleinik type estimate similar to the case when the diffusion is local. First we prove that the entropy solution can be obtained by adding a small viscous term $μu_{xx}$ and letting $μ\to 0$. Then, by using uniform Oleinik estimates for the viscous approximation we are able to prove the well-posedness of the entropy solutions with $L^1$-initial data. Using a scaling argument and hyperbolic estimates given by Oleinik's inequality, we obt…
We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a …
Numerical simulation of singularity propagation modeled by linear convection equations with spatially heterogeneous nonlocal interactions
该研究针对带空间异质性非局部相互作用的线性对流方程，解析推导了非局部区域奇异性传播的刻画方程，并用渐近相容格式离散开展数值模拟，揭示异质性与局部-非局部转换对奇异性传播的影响。
We study the existence and uniqueness of source-type solutions to the Cauchy problem for the heat equation with fast convection under certain tail control assumptions. We allow the solutions to change sign, but we will in fact show that they have the same sign as the initial data, which is a multiple of the Dirac delta. As an application, we obtain the large-time behaviour of nonnegative bounded solutions with integrable initial data to heat equations with fast convection, covering the case of several dimensions that remained open since the end of last century.
We consider a convection-diffusion model with linear fractional diffusion in the sub-critical range. We prove that the large time asymptotic behavior of the solution is given by the unique entropy solution of the convective part of the equation. The proof is based on suitable a-priori estimates, among which proving an Oleinik type inequality plays a key role.
Doubly nonlinear parabolic equation involving a mixed local-nonlocal operator and a convection term
本文研究了包含对流项与混合局部-非局部算子的双重退化抛物型方程，证明了弱-温和解的存在唯一性，并分析了其稳定化、灭绝与爆破行为对非线性条件的依赖。
On the decay estimates of a nonlocal convection-diffusion Hamer system
本文研究多维Hamer模型的双曲-椭圆耦合形式，在混合Besov空间中建立小初值全局适定性，放宽了已有正则性假设，还得到临界Besov空间下的最优时间衰减估计，并在零质量条件下推导了改进衰减速率。
In this paper we study the asymptotic behaviour of a nonlocal nonlinear parabolic equation governed by a parameter. After giving the existence of unique branch of solutions composed by stable solutions in stationary case, we gives for the parabolic problem $L^\infty $ estimates of solution based on using the Moser iterations and existence of global attractor. We finish our study by the issue of asymptotic behaviour in some cases when $t\to \infty$.
In this paper we study the asymptotic behaviour of a nonlocal nonlinear parabolic equation governed by a parameter. After giving the existence of unique branch of solutions composed by stable solutions in stationary case, we gives for the parabolic problem $L^\infty $ estimates of solution based on using the Moser iterations and existence of global attractor. We finish our study by the issue of asymptotic behaviour in some cases when $t\to \infty$.
In this paper we give a new compactness criterion in the Lebesgue spaces $L^p((0,T)\times Ω)$ and use it to obtain the first term in the asymptotic behaviour of the solutions of a nonlocal convection diffusion equation. We use previous results of Bourgain, Brezis and Mironescu to give a new criterion in the spirit of the Aubin-Lions-Simon Lemma.
In this note we study the behaviour as $s\to 0^+$ of some semigroup based Besov seminorms associated with a non-symmetric and hypoelliptic diffusion with a drift. Our results generalise a previous one of Maz'ya and Shaposhnikova for the classical fractional Sobolev spaces $W^{s,p}$, and they also underscore a new phenomenon caused by the presence of the drift.
Positivity and asymptotic behaviour of solutions to a generalized nonlocal fast diffusion equation
针对含类稳定算子与奇异非线性项的广义非局部快扩散方程，通过弱Harnack不等式分析解的正性并推导其渐近等价于标准分数阶快扩散方程的Barenblatt解，突破了算子一般性带来的方法限制。
Nonlocal Diffusion Elliptic System Modelling The Behaviour Of a Bacteria And a Living Nutrient
本文研究模拟细菌与活性养料行为的非局部椭圆系统，通过分支方法和隐函数定理证明了共存态正解的存在性与唯一性。
We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe flat boundary conditions. A Gamma-convergence result is discussed in the second …
We study a Robin boundary problem for degenerate parabolic equation. We suggest a notion of entropy solution and propose a result of existence and uniqueness. Numerical simulations illustrate some aspects of solution behaviour. Monodimensional experiments are presented.
We consider an evolution equation generalising the viscous Burgers equation supplemented by an initial condition which is a homogeneous random field. We develop a non-linear framework enabling us to show the existence and regularity of solutions as well as their long time behaviour.
Long-time behaviour of a nonlocal model for electroporation
研究非局部电穿孔模型的长期行为，证明了渐近自相似解的局部稳定性，通过分析一阶矩方程和与传输问题的比较，揭示了传输项对长期行为的驱动作用。
We study local (the heat equation) and nonlocal (convolution type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single …
We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or twice; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviours at the points where the diffusivity …
In this paper we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.
In these lecture notes, we address the problem of large-time asymptotic behaviour of the solutions to scalar convection-diffusion equations set in ${R}^N$. The large-time asymptotic behaviour of the solutions to many convection-diffusion equations is strongly linked with the behavior of the initial data at infinity. In fact, when the initial datum is integrable and of mass $M$, the solutions to the equations under consideration oftentimes behave like the associated self-similar profile of mass $M$, thus emphasising the role of scaling variables in these scenarios. However, these equations can …
Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the …
On a nonlocal two-phase flow with convective heat transfer
本文研究具有Caginalp型对流换热的非局部Cahn-Hilliard-Navier-Stokes两相流系统，证明了弱解的全局存在性，并证明当卷积核适当选取时非局部解收敛到局部模型。
We consider a diffusion model with limit cycle reaction functions, in the presence of convection. We select a set of functions derived from a realistic reaction model: the Schnakenberg equations. This resultant form is unsymmetrical. We find a transformation which maps the irregular equations into model form. Next we transform the dependent variables into polar form. From here, a travelling wave analysis is performed on the radial variable. Results are complex, but we make some simple estimates. We carry out numerical experiments to test our analysis. An initial `knock' starts the propagation …
We obtain upper bounds for the quenching time of the solutions of the nonlocal parabolic MEMS equation $u_t=Δu+\lam/(1-u)^2(1+χ\int_Ω1/(1-u) dx)^2$ in $Ω\times (0,\infty)$, $u=0$ on $\1Ω\times (0,\infty)$, $u(x,0)=u_0$ in $Ω$, when $λ$ is large. We prove the compactness of the quenching set under a mild condition on the initial data. When $Ω=B_R$ and $u_0$ is radially symmetric and monotone decreasing in $0\le r\le R$, we prove that the point $x=0$ is the only possible quenching set. When $u_0$ also satisfies some strict concavity assumption, we prove that for any $β\in (2,3)$ the solution sat…
Positivity and long-term behaviour of a diffusion model with measure-valued nonlocal reaction term
研究了在实线上具有非局部和奇异反应项的扩散方程解的行为，即由原点处的狄拉克源或汇给出。通过简化表示例如一个能够远距离感知浓度但干预原点的控制系统。正解（对于正初始条件）不能在所有参数设置下保证。我们确定了参数范围和正初始条件的单调性和对称性条件，使得可以得出解在所有时间的正性。此外，我们提供了确保系统收敛到常数稳态（点wise）的条件，该稳态位于观测区域之外。技术上，我们广泛使用拉普拉斯变换论证来获得这些结果。
Taking up a variational viewpoint, we present some nonlocal-to-local asymptotic results for various kinds of integral functionals. The content of the thesis comprises the contributions first appeared in some research papers in collaboration with J. Berendsen, A. Cesaroni, A. Chambolle, and M. Novaga.
Large time behaviour in the multidimensional nonlocal Fisher-KPP equation
本文研究多维非局部Fisher-KPP方程等值集的扩张，推导出渐近对数延迟及方向依赖的常数修正项，深化了Freidlin-Gärtner型公式的精度。
We obtain $L^q$--$L^p$ decay estimates, $1\le q<p<\infty$ for solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u=0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of Lévy type. We obtain these estimates in terms only of the behaviour of the kernel at infinity, without any information of its behaviour at the origin. This includes bounded and unbounded transition probability densities. An equivalence between the decay and a restricted Nash inequality is shown. We also prove that $\lim_{t\to \infty}\|u(t)\|_\infty=0$. Finally we deal w…
Long-time behaviour of a nonlocal stochastic fractional reaction--diffusion equation arising in tumour dynamics
本文研究了源于肿瘤动力学的非局部随机分数反应-扩散方程的长期行为，通过分析噪声对肿瘤扩散和灭绝的影响，揭示了异常扩散与分数噪声的相互作用机制。
The Oberbeck--Boussinesq approximation and Rayleigh--Benard convection revisited
本文研究非均匀温度边界下 Oberbeck--Boussinesq 近似，考虑非局部扰动项，并分析其初边值问题的长时间行为。
Local behaviour of the mixed local and nonlocal problems with nonstandard growth
本文研究了具有非标准增长条件 $1<p\le sq$ 的混合局部与非局部泛函极小子的局部行为，通过正性扩展方法证明了其局部有界性、局部 Hölder 连续性和 Harnack 不等式，将现有文献中 $p>sq$ 条件下的结果进行了自然推广。
Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang model of collective behaviour
本文研究具有矩阵值通信核的多维Euler-alignment系统，证明其与非局部Aw--Rascle--Zhang交通模型形式等价，并通过单一退化无压Navier--Stokes逼近建立两种表述的测度解全局存在性及弱-强唯一性原理，从而严格论证二者在无粘极限下的对应关系。
Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
In this paper we study nonlocal problems that are analogous to the local ones given by the Laplacian or the p-Laplacian with dynamical boundary conditions. We deal both with smooth and with singular kernels and show existence and uniqueness of solutions and study their asymptotic behaviour as t goes to infinity.
The existence and nonexistence for time nonlocal evolution equations with superlinear sources
研究具有超线性源的时间非局部发展方程初值问题，通过从属和单调迭代方法，分析其可解行为，建立存在性与不存在性充分条件，结果与初值指标相关且有临界维数现象，还通过例子展示适用性。
We show that the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains strongly differs from the one of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data. In this classical case it is known that, at least in a suitable weak sense, solutions of non-homogeneous Dirichlet problem are unique and tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that such results are false f…
We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $Ω\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal{L}$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $(-Δ)^s$ ($0<s<1$) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity $f$ is increasing and looks like a …
Let $Ω\subset\mathbb{R}^n$ be a $C^2$ bounded domain and $χ>0$ be a constant. We will prove the existence of constants $λ_N\geλ_N^{\ast}\geλ^{\ast}(1+χ\int_Ω\frac{dx}{1-w_{\ast}})^2$ for the nonlocal MEMS equation $-Δv=\lam/(1-v)^2(1+χ\int_Ω1/(1-v)dx)^2$ in $Ω$, $v=0$ on $\1Ω$, such that a solution exists for any $0\leλ<λ_N^{\ast}$ and no solution exists for any $λ>λ_N$ where $λ^{\ast}$ is the pull-in voltage and $w_{\ast}$ is the limit of the minimal solution of $-Δv=\lam/(1-v)^2$ in $Ω$ with $v=0$ on $\1Ω$ as $λ\nearrow λ^{\ast}$. We will prove the existence, uniqueness and asymptotic behavi…
We prove convergence of positive solutions to \[ u_t = uΔu + u\int_Ω |\nabla u|^2, \qquad u\rvert_{\partialΩ} =0, \qquad u(\cdot,0)=u_0 \] in a bounded domain $Ω\subset \mathbb{R}^n$, $n\ge 1$, with smooth boundary in the case of $\int_Ωu_0=1$ and identify the $W_0^{1,2}(Ω)$-limit of $u(t)$ as $t\to \infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\int_Ωu_0<1$ and $\int_Ωu_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of $\int_Ω |\nabla u…
We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions' concentration-compactness principle, we prove that either an optimal shape exists, or there exists a minimizing sequence consisting of two "pieces" whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.
Spectral theory of plasmonic resonances in the nonlocal regime
本文基于非局域介质的非局域 regime，利用流体动力学 Drude 模型的准静态近似，推导出描述共振行为的分析 Fredholm 运算符，揭示非局域模型具有离散实谱结构，改变传统局部理论中无限多个表面等离子体模的结论。
The aim of this paper is to provide a comprehensive study of some linear nonlocal diffusion problems in metric measure spaces. These include, for example, open subsets in $\mathbb{R}^N$, graphs, manifolds, multi-structures or some fractal sets. For this, we study regularity, compactness, positiveness and the spectrum of the stationary nonlocal operator. Then we study the solutions of linear evolution nonlocal diffusion problems, with emphasis in similarities and differences with the standard heat equation in smooth domains. In particular prove weak and strong maximum principles and describe th…
On the precise cusped behaviour of extreme solutions to Whitham-type equations
本文证明了两类非局部方程（包括单向和双向Whitham方程的最高稳态尖峰波）非平凡解在原点的精确首阶渐近行为，部分解决了相关猜想，且方法可推广至其他非局部方程并确定任意阶导数的渐近行为。
Fredholm Property of Nonlocal Problems for Integro-Differential Hyperbolic Systems
本文研究了非局部时间周期边值问题的Fredholm性质，证明了在特定正则性条件下，此类问题在连续和时间周期函数空间中满足Fredholm替代定理。
Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential
本文研究带Hardy位势的混合局部-非局部椭圆方程，通过变换建立解的一致渐近估计，分情形证明正解存在性，并在次线性小扰动下得到多解结果。
Uniqueness for local-nonlocal elliptic equations
本文研究带变系数ρ的混合局部-非局部椭圆方程，在关于ρ无穷远处行为的最优假设下证明加权Lebesgue空间中解的唯一性，并指出假设失效时会出现非唯一性，同时考察了相应的抛物型方程。
In this paper, we first consider two scalar nonlocal diffusion problems with a free boundary and a fixed boundary. We obtain the global existence, uniqueness and longtime behaviour of solution of these two problems. The spreading-vanishing dichotomy and sharp criteria for spreading and vanishing are established. We also prove that accelerated spreading could happen if and only if a threshold condition is violated by kernel function. Then we discuss a classical Lotka-Volterra predator-prey model with nonlocal diffusions and a free boundary which can be seen as nonlocal diffusion counterpart of …
We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the behaviour of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffus…
On numerical methods and error estimates for degenerate fractional convection-diffusion equations
针对一大类非线性非局部可能退化的对流扩散方程，提出了一种收敛的数值方法，并开发了新的Kuznetsov型理论，得到了主导数阶数在1到2之间的最优误差估计。
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…
We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem: \begin{align*} \left\{\begin{array}{l l} { d(-Δ)^{s}u+ u= \abs{u}^{p-1}u } \text{ in $Ω,$ } { \mathcal{N}_{s}u=0 } \text{ in $\mathbb{R}^{n}\setminus \overlineΩ,$} {u>0} \text{ in $Ω,$} \end{array} \right.\end{align*} where $Ω\subset \mathbb{R}^{n}$ is a bounded domain of class $C^{1,1}$, $1<p<\frac{n+s}{n-s},\,n> \max \left\{1, 2s \right\}, 0<s<1,\,d>0$ and $\mathcal{N}_{s}u$ is the nonlocal Neumann derivative. We show that for small $d,$ the least energy solutions $u_d$ of the above…
In this paper, we examine the long-time dynamics of an epidemic model whose diffusion and reaction terms involve nonlocal effects described by suitable convolution operators.The spreading front of the disease is represented by the free boundaries in the model. We show that the model is well-posed, its long-time dynamical behaviour is characterised by a spreading-vanishing dichotomy, and we also obtain sharp criteria to determine the dichotomy. Some of the nonlocal effects in the model pose extra difficulties in the mathematical treatment, which are dealt with by introducing new approaches. The…
In the present work we deal with the existence of solutions for optimal control problems associated to transport equations. The behaviour of a population of individuals will be influenced by the presence of a population of control agents whose role is to lead the dynamics of the individuals towards a specific goal. The dynamics of the population of individuals is described by a suitable nonlocal transport equation, while the role of the population of agents is designed by the optimal control problem. This model has been first studied in [12] for a class of continuous nonlocal potentials, while…
We introduce and analyze a nonlocal version of the one-phase Stefan problem in which, as in the classical model, the rate of growth of the volume of the liquid phase is proportional to the rate at which energy is lost through the interphase. We prove existence and uniqueness for the problem posed on the line, and on the half-line with constant Dirichlet data, and in the radial case in several dimensions. We also describe the asymptotic behaviour of both the solution and its free boundary. The model may be of interest to describe the spreading of populations in hostile environments.
We are concerned with the asymptotic behaviour of classical solutions of systems of the form u_t = Au_xx + f(u, u_x), x in R, t>0, u(x,t) a vector in RN, with u(x,0)= U(x), where A is a positive-definite diagonal matrix and f is a 'bistable' nonlinearity satisfying conditions which guarantee the existence of a comparison principle. Suppose that there is a travelling-front solution w with velocity c, that connects two stable equilibria of f. We show that if U is bounded, uniformly continuously differentiable and such that w(x) - U(x) is small when the modulus of x is large, then there exists y …
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。