Immanuel Zachhuber
We prove finite speed of propagation for the multiplicative stochastic wave equation in two and three dimensions which leads us to a global space-time well-posedness result for the cubic nonlinear equation in the analogue of the energy space.
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We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.
Quantitative propagation of chaos for particle systems with bounded kernels and multiplicative noise
研究在漂移和扩散系数中均存在相互作用的随机粒子系统，证明在漂移核有界的情况下，定量混沌传播的成立，扩展了Jabin和Wang的指数大数定律到扩散核，并处理熵演化方程中乘性噪声带来的误差项。
On blow up NLS with a multiplicative noise
研究乘性噪声对NLS爆破速度的影响，证明噪声导致爆破的概率较小，并给出大偏差上界。
We give an example of a stochastic Hamilton-Jacobi equation $du = H(Du) dξ$ which has an infinite speed of propagation as soon as the driving signal $ξ$ is not of bounded variation.
The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix dependent of the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentration. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobo…
We prove finite speed of propagation for stochastic porous media equations perturbed by linear multiplicative space-time rough signals. Explicit and optimal estimates for the speed of propagation are given. The result applies to any continuous driving signal, thus including fractional Brownian motion for all Hurst parameters. The explicit estimates are then used to prove that the corresponding random attractor has infinite fractal dimension.
We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant driving term and random nonlinearity to model the influence of the obstacle field. For the case of isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof consists of a discretization of the evolution equation…
We characterize the asymptotic speed of propagation of almost planar solutions to a semilinear viscous parabolic equation, with periodic nonlinearity.
This note is concerned with proving the finite speed of propagation for some non-local porous medium equation by adapting arguments developed by Caffarelli and Vázquez (2010).
We investigate multidimensional model for incompressible non-Newtonian fluids. Using method of energy estimates we prove the property of finite speed of propagations of the solution support for this problem. We find sharp bounds of the propagations by $L^2$--norm and $L^1$--norm of initial data.
We show that a double degenerate thin film equation, which originated from modeling of viscous coating flow on a spherical surface, has finite speed of propagation for nonnegative strong solutions and hence there exists an interface or free boundary separating the regions where solution $u>0$ and $u=0$. Using local entropy estimates we also obtain an upper bound for the rate of the interface propagation.
Finite speed of propagation is established for non-negative weak solutions to a thin film approximation of the two-phase Muskat problem. The temporal expansion rate of the support matches the scale invariance of the system. Moreover, we determine sufficient conditions on the initial data for the occurrence of waiting time phenomena.
Diffusion in porous media with hysteresis and bounded speed of propagation
本文研究多孔介质中水分传播问题，在质量通量与压力梯度非线性关系及毛细压力滞后情形下，证明了有界传播速度，并给出解的存在唯一性条件及速度上界。
From Stochastic Zakharov System to Multiplicative Stochastic Nonlinear Schr{ö}dinger Equation
研究时间白噪声驱动、空间有色噪声的 Zakharov 系统在离子声速趋于无穷时向乘性随机非线性 Schr{ö}dinger 方程的收敛性，通过扩散近似框架与扰动测试函数方法克服噪声导致的奇异性，并证明一维情形下解的存在唯一性与依概率收敛。
We prove that strongly continuous groups generated by first order systems on Riemannian manifolds have finite propagation speed. Our procedure provides a new direct proof for self-adjoint systems, and allows an extension to operators on metric measure spaces. As an application, we present a new approach to the weak Huygens' principle for second order hyperbolic equations.
We propose some modification of Maxwell's equations describing mediums which electric and magnetic properties are changed essentially after interaction with outer electromagnetic field. We show for such mediums that electromagnetic waves have finite speed of propagations property for some time depending on initial energy of electromagnetic field and nonlinear parameters of the problem which are responsible for properties of medium.
We consider a degenerate abstract wave equation with a time-dependent propagation speed. We investigate the influence of a strong dissipation, namely a friction term that depends on a power of the elastic operator. We discover a threshold effect. If the propagation speed is regular enough, then the damping prevails, and therefore the initial value problem is well-posed in Sobolev spaces. Solutions also exhibit a regularizing effect analogous to parabolic problems. As expected, the stronger is the damping, the lower is the required regularity. On the contrary, if the propagation speed is not re…
The peridynamic analogue of the wave equation does not have finite speed propogation. We show, for one dimensional linear peridynamics, that solutions do nonetheless satisfy estimates analogous to those satisfied by solutions of the wave equations. More precisely, the solution and all derivatives become small as we go away from the domain of dependence of the initial data.
Finite propagation speed properties in mathematical elastic and viscoelastic models are fundamental in many applications where the data exhibits propagating fronts. We note particularly that this property is observed in biomechanical imaging of tissue, in particular in the supersonic imaging experiment, and also in geophysics and ocean acoustics. With these applications in mind, noting that there are many other applications as well, we present finite propagation speed results for very general integro-differential, anisotropic, viscoelastic linear models, which are not necessarily of convolutio…
Discretization is a fundamental step in numerical analysis for the problems described by differential equations, and the difference between the continuous model and discrete model is one of the most important problems. In this paper, we consider the difference in the effect of the time-dependent propagation speed on the energy estimate of the solutions for the wave equation and the semi-discrete wave equation which is a discretization with respect to space variables.
We investigate the long time behaviour of the $L^2$-energy of solutions to wave equations with variable speed. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property.
Existence and finite speed of propagation of solutions for a multi-dimensional fractional thin-film equation
本文研究有界域上带Neumann边界条件的谱分数阶Laplacian薄膜方程初边值问题，在强滑移完全润湿条件下通过局部熵估计证明了解的存在性、支撑集有限传播速度及等待时间下界。
We consider an abstract wave equation with a propagation speed that depends only on time. We investigate well-posedness results with finite derivative loss in the case where the propagation speed is smooth for positive times, but potentially singular at the initial time. We prove that solutions exhibit a finite derivative loss under a family of conditions that involve the blow up rate of the first and second derivative of the propagation speed, in the spirit that the weaker is the requirement on the first derivative, the stronger is the requirement on the second derivative. Our family of condi…
On the propagation speed of the single monostable equation
本文研究单稳型反应扩散方程的速度选择问题，通过分析最小行波前锋衰减速率提出充分必要条件，且该基于比较原理的论证可推广至非局部扩散方程等更一般的单稳动力系统。
We study a porous medium equation with fractional potential pressure: $$ \partial_t u= \nabla \cdot (u^{m-1} \nabla p), \quad p=(-Δ)^{-s}u, $$ for $m>1$, $0<s<1$ and $u(x,t)\ge 0$. The problem is posed for $x\in \mathbb{R}^N$, $N\geq 1$, and $t>0$. The initial data $u(x,0)$ is assumed to be a bounded function with compact support or fast decay at infinity. We establish existence of a class of weak solutions for which we determine whether the property of compact support is conserved in time depending on the parameter $m$, starting from the result of finite propagation known for $m=2$. We find t…
From the finite difference method for wave equation on p.c.f. fractals, we would expect that infinite prorogation speed property for wave solutions on a large class of p.c.f. fractals. We prove that is true if the heat kernel satisfies the sub-Gaussian lower bound. Furthermore, we provide a sub-Gaussian upper bound for wave kernel given the heat kernel sub-Gaussian upper bound.
We investigate an abstract wave equation with a time-dependent propagation speed, and we consider both the non-dissipative case, and the case with a strong damping that depends on a power of the elastic operator. Previous results show that, depending on the values of the parameters and on the time regularity of the propagation speed, this equation exhibits either well-posedness in Sobolev spaces, or well-posedness in Gevrey spaces, or ill-posedness with severe derivative loss. In this paper we examine some critical cases that were left open by the previous literature, and we show that they fal…
In this paper we consider a model for the diffusion of a population in a strip-shaped field, where the growth of the species is governed by a Fisher-KPP equation and which is bounded on one side by a road where the species can have a different diffusion coefficient. Dirichlet homogeneous boundary conditions are imposed on the other side of the strip. We prove the existence of an asymptotic speed of propagation which is greater than the one of the case without road and study its behavior for small and large diffusions on the road. Finally we prove that, when the width of the strip goes to infin…
This paper is devoted to the study of the asymptotic behaviors of the minimal speed of propagation of pulsating traveling fronts solving the Fisher-KPP reaction-advection-diffusion equation within either a large drift, a mixture of large drift and small reaction, or a mixture of large drift and large diffusion. We consider a periodic heterogenous framework and we use the formula of Berestycki, Hamel and Nadirashvili \cite{bhn2} for the minimal speed of propagation to prove the asymptotics in any space dimension $N.$ We express the limits as the maxima of certain variational quantities over the…
For the incompressible and the isentropic compressible Euler equations in arbitrary space dimension, we establish the principle of localised relative energy, thus generalising the well-known relative energy method. To this end, we adapt classical arguments of C. Dafermos to the Euler equations. We give several applications to the behaviour of weak solutions, like local weak-strong uniqueness, local preservation of smoothness, and finite speed of propagation for the isentropic system.
We show that the boundary conditions entering in the definition of the self-adjoint operator describing the Laplacian plus a finite number of point interactions are local if and only if the corresponding wave equation has finite speed of propagation
Appearance of Strauss-type exponent in semilinear wave equations with time-dependent speed of propagation
本文研究带时变传播速度与散射阻尼的半线性波动方程的小初值爆破现象，采用带Liouville–Green近似特殊解守恒量的检验函数法，得到次Strauss指数下的爆破结果，覆盖半线性广义Tricomi方程。
We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming the initial datum is localized with respect to a coordinate having slow diffusion rate, we bound the corresponding directional velocity of the support along the flow. The expansion rate is shown to be optimal for large times.
We study a porous medium equation with fractional potential pressure: $$ \partial_t u= \nabla \cdot (u^{m-1} \nabla p), \quad p=(-Δ)^{-s}u, $$ for $m>1$, $0<s<1$ and $u(x,t)\ge 0$. To be specific, the problem is posed for $x\in \mathbb{R}^N$, $N\geq 1$, and $t>0$. The initial data $u(x,0)$ is assumed to be a bounded function with compact support or fast decay at infinity. We establish existence of a class of weak solutions for which we determine whether, depending on the parameter $m$, the property of compact support is conserved in time or not, starting from the result of finite propagation k…
In the present paper, we study the Cauchy problem for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds. Our aim of this paper is to prove a small data blow-up result and an upper estimate of lifespan of the problem for a suitable compactly supported initial data in the subcritical and critical cases of the Strauss type. The proof is based on the framework of the argument in the paper [17]. One of our new contributions is to construct two families of special solutions to the free equation (see (2.16) or (2.18) as the test functions and prove their …
Generalized energy conservation for linear wave equations with time-dependent propagation speed
研究具有时变传播速度的波动方程的广义能量守恒条件，通过分析Fourier分量增长推导能量上界，并用反例证明其最优性。
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 study the long time behavior of energy solutions for a class of wave equation with time-dependent mass and speed of pro\-pagation. We introduce a classification of the potential term, which clarifies whether the solution behaves like the solution to the wave equation or Klein-Gordon equation. Moreover, $L^q-L^2, q\in [1, 2]$ estimates for scale-invariant models are derived and applied to obtain global in time small data energy solutions for the semilinear Klein-Gordon equation in anti de Sitter spacetime.
We prove a number of results on the geometry associated to the solutions of evolution equations given by first-order differential operators on manifolds. In particular, we consider distance functions associated to a first-order operator, and discuss the associated geometry, which is sometimes surprisingly different to riemannian geometry.
Existence of solutions of semilinear wave equations with time-dependent propagation speed and time derivative nonlinearity
本文研究含时变传播速度和时间导数非线性项的半线性波动方程，给出传播速度的可积性条件，证明膨胀宇宙中解的全局存在性与寿命下界，同时证明收缩宇宙中解的有限时间爆破并给出寿命上界。
The speeds of propagation for the monostable Lotka-Volterra competition-diffusion system in general unbounded domains
研究单稳Lotka-Volterra竞争扩散系统在无界域中传播速度的定义与性质，推导了扩散能力相同且相互作用较弱时的传播速度上界，并在不同域结构下获得上下界。
We prove maximal speed estimates for nonlinear quantum propagation in the context of the Hartree equation. More precisely, under some regularity and integrability assumptions on the pair (convolution) potential, we construct a set of energy and space localized initial conditions such that, up to time-decaying tails, solutions starting in this set stay within the light cone of the corresponding initial datum. We quantify precisely the light cone speed, and hence the speed of nonlinear propagation, in terms of the momentum of the initial state.
We consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. Under the null condition for such systems, the global existence of small amplitude solutions is known. In this paper, we will show that the global solution is asymptotically free in the energy sense, by obtaining the asymptotic pointwise behavior of the derivatives of the solution. Nonetheless we can also show that the pointwise behavior of the solution itself may be quite different from that of the free solution. In connection with the above results, a theorem is …
We are concerned with the nonlinear problem $u_t=u_{xx}+f(u)$, where $f$ is of combustion type, coupled with the Stefan-type free boundary $h(t)$. According to [4,5], for some critical initial data, the transition solution $u$ locally uniformly converges to $θ$, which is the ignition temperature of $f$, and the free boundary satisfies $h(t)=C\sqrt{t}+o(1)\sqrt{t}$ for some positive constant $C$ and all large time $t$. In this paper, making use of two different approaches, we establish more accurate upper and lower bound estimates on $h(t)$ for the transition solution, which suggest that the no…
We prove that asymptotic global consensus is always reached in the Hegselmann-Krause model with finite speed of information propagation $\mathfrak{c}>0$ under minimal (i.e., necessary) assumptions on the influence function. In particular, we assume that the influence function is globally positive, which is necessary for reaching global consensus, and such that the agents move with speeds strictly less than $\mathfrak{c}$, which is necessary for well-posedness of solutions. From this point of view, our result is optimal. The proof is based on the fact that the state-dependent delay, induced by …
We consider the Cauchy problem of systems of quasilinear wave equations in 2-dimensional space. We assume that the propagation speeds are distinct and that the nonlinearities contain quadratic and cubic terms of the first and second order derivatives of the solution. We know that if the all quadratic and cubic terms of nonlinearities satisfy $Strong$ $Null$-$condition$, then there exists a global solution for sufficiently small initial data. In this paper, we study about the lifespan of the smooth solution, when the cubic terms in the quasi-linear nonlinearities do not satisfy the Strong null-…
There exists a critical speed of propagation of the line solitons in the Zakharov-Kuznetsov (ZK) equation such that small transversely periodic perturbations are unstable for line solitons with larger-than-critical speeds and orbitally stable for those with smaller-than-critical speeds. The normal form for transverse instability of the line soliton with a nearly critical speed of propagation is derived by means of symplectic projections and near-identity transformations. Justification of this normal form is provided with the energy method. The normal form predicts a transformation of the unsta…
In this paper, we introduce a finite propagation speed perturbation of the incompressible Navier-Stokes equations (NS). The model we consider is inspired by a hyperbolic perturbation of the heat equation due to Cattaneo (Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ., Modena, 1949, 3:83-101) and by an equation that Vishik and Fursikov (Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1977, 4:531-576) investigated in order to find statistical solutions to (NS). We prove that the solutions to …
In this paper, we analyse propagating fronts in the context of hyperbolic theories of dissipative processes. These can be considered as a natural alternative to the more classical parabolic models. Emphasis is given toward the numerical computation of the invasion velocity.
We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We …
Acceleration or finite speed propagation in integro-differential equations with logarithmic Allee effect
该研究针对带弱退化非线性项的积分微分方程传播问题，分析扩散核尾部对扩张速率的影响，得到行波存在性及传播模式的精确分界与加速速率，方法通用且有数值验证。
On $α$-entropy solutions of a nonlocal thin film equation: existence and finite speed of propagation
本文首次提出非局部薄片方程的α-熵估计，证明了有限传播速度及等待时间现象，揭示了非局部效应与经典薄片动力学的相互作用。
In this paper we focus on three problems about the spreading speeds of nonlocal dispersal Fisher-KPP equations. First, we study the signs of spreading speeds and find that they are determined by the asymmetry level of the nonlocal dispersal and $f'(0)$, where $f$ is the reaction function. This indicates that asymmetric dispersal can influence the spatial dynamics in three aspects: it can determine the spatial propagation directions of solutions, influence the stability of equilibrium states, and affect the monotone property of solutions. Second, we give an improved proof of the spreading speed…
This paper is devoted to the study of propagation phenomena for a Lotka-Volterra reaction-advection-diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trival steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reacti…
We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed $\mathfrak{c}>0$. This leads to a system of functional differential equations with state-dependent delay. We prove that, if initially the agents travel slower than $c$, then the discrete model admits unique global solutions. Moreover, under a generic assumption on the influence function, we show that there exists a critical information propagation speed $\mathfrak{c}^\ast>0$ such that if $\mathfrak{c}\geq\mathfrak{c}^\ast$, the system exhibits asymptotic flocking in the sense of the cla…
Propagation speed of traveling waves for diffusive Lotka-Volterra system with strong competition
本研究通过比较论证建立了强竞争扩散Lotka-Volterra系统中双稳行波传播速度符号的充分条件，揭示对称情形下快扩散者优势范围更广，且竞争系数差异大时结果不可逆转。
We study the time asymptotic propagation of solutions to the reaction-diffusion cooperative systems with fractional diffusion. We prove that the propagation speed is exponential in time, and we find the precise exponent of propagation. This exponent depends on the smallest index of the fractional laplacians and on the principal eigenvalue of the matrix DF(0) where F is the reaction term. We also note that this speed does not depend on the space direction.
Linear vs. nonlinear speed selection of the front propagation into unstable states
本文针对Lotka-Volterra竞争系统，提出速度选择的充分必要条件，并揭示单稳系统线性选择的本质。
This paper studies the phenomenon of invasion for heterogeneous reaction-diffusion equations in periodic domains with monostable and combustion reaction terms. We give an answer to a question rised by Berestycki, Hamel and Nadirashvili in [5] concerning the connection between the speed of invasion and the speed of fronts. To do so, we extend the classical Freidlin-Gartner formula to such equations, using a geometrical argument devised by Rossi in [17], and derive some bounds on the speed of fronts using estimates on the heat kernel.
In this paper we investigate the dynamical properties of a spatially periodic reaction-diffusion system {whose reaction terms are of hybrid nature in the sense that they are partly competitive and partly cooperative depending on the value of the solution. This class of problems includes various biologically relevant models and in particular many models focusing on the Darwinian evolution of species. We start by studying the principal eigenvalue of the associated differential operator and establishing a minimal speed formula for linear monotone systems. In particular, we show that the generaliz…
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