In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e.$\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-γ}*v^2)v$ with $γ\in (0,n)$, where $v=v(x,t)$ is an unknown function on $\mathbb{R}^n\times[0,T)$. Our aim of the present paper is to prove a small data blow-up result and show an upper estimate of lifespan of the problem for slowly decaying positive initial data $(v(x,0),\partial_t v(x,0))$ such as $\partial_t v(x,0)=O(|x|^{-(1+ν)})$ as $|x|\rightarrow\infty$. Here $ν$ belongs to the scaling supercritical case $ν<\frac{n-γ}{2}$. Our main new contribution is to estimate the convolution term in high spatial dimensions, i.e. $n\ge 4$. This paper is the first blow-up result to treat wave equations with the cubic convolution in high spatial dimensions ($n\ge 4$).
For the one-dimensional case, we establish the long-time asymptotics of solution to Cauchy problem and prove existence of modified wave operators. In particular, we show that the part of the wave travels ballistically if the potential is square summable and this result is sharp.
Local time decay for fractional Schrödinger operators with slowly decaying potentials and a weaker Agmon type estimate in a classically forbidden region
In this short note, we prove Strichartz estimates for Schrödinger operators with slowly decaying singular potentials in dimension two. This is a generalization of the recent results by Mizutani, which are stated for dimension greater than two. The main ingredient of the proof is a variant of Kato's smoothing estimate with a singular weight.
The initial value problem for the homogeneous Schrödinger equation is investigated for radially symmetric initial data with slow decay rates and not too wild oscillations. Our global wellposedness results apply to initial data for which Strichartz estimates fail.
In this paper, kinds of Schrödinger type equations with slowly decaying linear potential and power type or convolution type nonlinearities are considered. By using the concentration compactness principle, the sharp Gagliardo-Nirenberg inequality and a refined estimate of the linear operator, the existence and orbital stability of standing waves in $L^2$-subcritical and $L^2$-critical cases are established in a systematic way.
In this paper, we study the spatial propagation dynamics of a parabolic-elliptic chemotaxis system with logistic source which reduces to the well-known Fisher-KPP equation without chemotaxis. It is known that for fast decaying initial functions, this system has a finite spreading speed. For slowly decaying initial functions, we show that the accelerating propagation will occur and chemotaxis does not affect the propagation mode determined by slowly decaying initial functions if the logistic damping is strong, that is, the system has the same upper and lower bounds of the accelerating propagati…
For Schrödinger equations with a class of slowly decaying repulsive potentials, we show that the solution satisfies global-in-time Strichartz estimates for any admissible pairs. Our admissible class of potentials includes the positive homogeneous potential $Z|x|^{-μ}$ with $Z>0$ and $0<μ<2$ in three and higher dimensions, especially the repulsive Coulomb potential. The proof employs several techniques from scattering theory such as the long time parametrix construction of Isozaki-Kitada type, propagation estimates and local decay estimates.
Consider the Cauchy problem of incompressible Navier-Stokes equations in $\mathbb{R}^3$ with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the existence of a time-global weak solution has been known. However, such data do not include constants, and the only known global solutions for non-decaying data are either for perturbations of constants, or when the velocity gradients are in $L^p$ with finite $p$. In this paper, we construct global weak solutions for non-decaying initial data whose local osci…
This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher-KPP reaction-diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time…
In Kenig and Toro's two-phase free boundary problem, one studies how the regularity of the Radon-Nikodym derivative $h= dω^-/dω^+$ of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that $\log h \in C^{0,α}(\partial Ω)$ implies that pointwise the boundary has a unique blow-up, which is the zero set of a homogeneous harmonic polynomial. In this note, we give examples of domains with $\log h \in C(\partial Ω)$ whose boundaries have points with non-unique blow-ups. Philosophically the examples arise from oscillating or rotating a blow…
In this paper, we consider a 3d cubic focusing nonlinear Schrödinger equation (NLS) with slowing decaying potentials. Adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a condition for scattering. It is actually sharp in some sense since the solution will blow up if it's false. The proof of blow-up part relies on the method of Du-Wu-Zhang \cite{DWZ}
In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curvature. Here we show that the decay of Hong $t^{-2-δ/2}$ where $δ\in(0,4)$ suffices.
We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the damped wave equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.
We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {\it fast} at infinity, in the sense that $ρ(x)\sim |x|^{-q}$ as $|x|\to +\infty$ with $q\ge 2.$ In the case when $q=2$, if $p$ is bigger than $m$, we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when $q>2$, we show that existence of global in time solutions always prevails. The case of …
The goal of this paper is to provide an algorithm that, for any sufficiently localised, divergence-free small initial data, explicitly constructs a localised external force leading to a rapidly dissipative solutions of the Navier-Stokes equations $\mathbb{R}^n$: namely, the energy decay rate of the flow will be forced to satisfy $\|u(t)\|_2^2 = o(t^{-(n+2)/2})$ as $t \to \infty$, which is beyond the usual optimal rate. An important feature of our construction is that this force can always be taken compactly supported in space-time, and its profile arbitrarily prescribed up to a spatial rescali…
The Fisher-KPP equation is a model for population dynamics that has generated a huge amount of interest since its introduction in 1937. The speed with which a population spreads has been computed quite precisely when the initial data decays exponentially. More recently, though, the case when the initial data decays more slowly has been studied. Building on the results of Hamel and Roques '10, in this paper we improve their precision for a broader class of initial data and for a broader class of equations. In particular, our approach yields the explicit highest order term in the location of the…
We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and "essentially Gaussian" coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the quantitative decorrelation on larger scales) correctly. The result is based on estimates on the Malliavin derivative for certain functionals which are basically averages of the gradient of the corr…
This paper is concerned with weighted energy estimates for solutions to wave equation $\partial_t^2u-Δu + a(x)\partial_tu=0$ with space-dependent damping term $a(x)=|x|^{-α}$ $(α\in [0,1))$ in an exterior domain $Ω$ having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polymonials are given and these decay rate are almost sharp, even when the initial data do not have compact support in $Ω$. The crucial idea is to use special solution of $\partial_t u=|x|^αΔu$ including Kummer's confluent hypergeometric functions.
The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundamental forms. It is shown that if the Gauss curvature satisfies an integrability condition, the surface has a global smooth isometric immersion in the three-dimensional Euclidean space even if the Gauss curvature decays very slowly at infinity. The new idea of the proof is based o…
We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term $ρ(x) u^p$ with $p>1$; this is a mathematical model of a thermal evolution of a heated plasma (see [25]). The density decays slowly at infinity, in the sense that $ρ(x)\lesssim |x|^{-q}$ as $|x|\to +\infty$ with $q\in [0, 2).$ We show that for large enough initial data, solutions blow-up in finite time for any $p>1$. On the other hand, if the initial datum is small enough and $p>\bar p$, for a suitable $\b…
Slowly travelling infinite point blow-up for the critical generalized KdV equation
Laboratoire de mathématiques de Versailles（凡尔赛数学实验室）；Université de Versailles St-Quentin en Y.（圣康坦凡尔赛大学）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）
We prove that, in a class of spherically symmetric spacetimes exhibiting stable trapping of null geodesics, linear waves cannot (uniformly) decay faster than logarithmically. When these linear waves are treated as a model for nonlinear perturbations, this slow decay is highly suggestive of nonlinear instability. We also prove that, in a large class of asymptotically flat, spherically symmetric spacetimes, logarithmic decay actually holds as a uniform upper bound. In the presence of stable trapping, this result is therefore the best one can obtain. In addition, we provide an application of thes…
We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we have to make the additional assumption that the initial data is Hölder continuous. Our hypotheses are much weaker, in terms of regularity and decay, than previous large-data well-posedness results …
We show new local $L^p$-smoothing estimates for the Schrödinger equation with initial data in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of solutions with initial data in modulation and $L^p$-spaces. The examples show sharpness of the smoothing estimates up to the endpoint regularity in a certain range. Moreover, the examples rule out global Strichartz estimates for initial data in $L^p(\mathbb{R}^d)$ for $d \geq 1$ and $p > 2$, which was previously known for $d \geq 2$. The estimates are applied to …
In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial_t^2u-Δu + \partial_tu = f(u) & \text{in}\ Ω\times (0,T), \\ u=0 & \text{on}\ \partialΩ\times (0,T), \\ u(0)=u_0, \partial_tu(0)=u_1 & \text{in}\ Ω, \end{cases} \end{equation*} in an exterior domain $Ω$ in $\mathbb{R}^N$ $(N\geq 2)$, where $f:\mathbb{R}\to \mathbb{R}$ is a smooth function behaves like $f(u)\sim |u|^p$. From the view point of weighted energy estimates given by Sobajima--Wakasugi \cite{SoWa4}, the existence of global-in-time solutions wi…
The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly s…
We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential induces growth in the linearized problem, however, solutions that start out small may blow-up in finite time.
The aim of the present paper is twofold:(1) We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators writing as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another-smaller or larger-Banach space under the condition…
We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - Δu + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^β) \times (H^{s-1} \cap L^r)$ with $r \in (1,2]$, $s\ge 0$, and $β= (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the …
We prove that 3-dimensional Schrodinger operator with slowly decaying potential has an absolutely continuous spectrum that fills the positive half-line. The asymptotics of Green's function is obtained as well.
Under the assumption that the initial velocity and outflow velocity are analytic in the horizontal variable, the local well-posedness of the geophysical boundary layer problem is obtained by using energy method in the weighted Chemin-Lerner spaces. Moreover, when the initial velocity and outflow velocity satisfy certain condition on a transversal plane, it is proved that the $W^{1,\infty}-$norm of any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem blows up in a finite time.
In this paper we consider the Cauchy problem for the 3D Navier-Stokes equations for incompressible flows. The initial data are assumed to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solutions can develop singularities in finite time. Assuming the maximum interval of existence to be finite, we give a unified discussion of various known solution properties as time approaches the blow-up time.
Consider the equation u_t=Δu-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=ϕ(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\fracω{|x|^2}$ as $|x|\to\infty$, for some $ω\neq0$, and $a(x)$ is on the order $|x|^m$ as $|x|\to\infty$, for some $m\in (-\infty,\infty)$. A solution to the above equation is called global if $T=\infty$. Under some additional technical conditions, we calculate a critical exponent $p^*$ such that global solutions exist for $p>p^*$, while for $1<p\le p^*$, all solutions blow up in finite time. We also show that when $V\equiv0$, the blow-up/global s…
This paper considers the immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations. The viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., $μ=\barμ θ^α, ~ λ=\barλ θ^α,~ κ=\barκ θ^α$, for constants $0\leq α\leq 1/(γ-1)$, $\barμ>0,~2\barμ+n\barλ\geq0,~\barκ\geq 0$, and adiabatic exponent $γ>1$). With prescribed decaying rate of the initial density across the vacuum boundary, we prove that: (1…
We prove that 3-dimensional Schrodinger operator with slowly decaying sparse potential has an a.c. spectrum that fills positive half-line. A new kind of WKB asymptotics for Green's function is found. The absense of positive eigenvalues is established as well.
Generically sharp decay and blowing up at infinity for a weak null wave system
Southern University of Science and Technology（南方科技大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Xi’an Jiaotong University（西安交通大学）
We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation $Ω$. First we study the behavior of the ground state when the coupling constant approaches $a\_*$ , the critical strength of the cubic nonlinearity for the focusing nonlinear Schr{ö}dinger equation. We prove that blow-up always happens at the center of the trap, with the blow-up profile given by the Gagliardo-Nirenberg solution. In particular, the blow-up scenario is independent of $Ω$, to leading order. …
In this paper, we are considering the Cauchy problem of the nonlinear heat equation $u\_t -Δu= u^{3 },\ u(0,x)=u\_0$. After extending Y. Meyer's result establishing the existence of global solutions, under a smallness condition of the initial data in the homogeneous Besov spaces $\dot{B}\_{p}^{-σ, \infty}(\mathbb{R}^{3})$, where $3 \textless{} p \textless{} 9$ and $σ=1-3/p$, we prove that initial data $u\_0\in \mathcal{S}(\mathbb{R}^{3})$, arbitrarily small in ${\dot B^{-2/3,\infty}\_{9}}(\mathbb{R}^{3})$, can produce solutions that explode in finite time. In addition, the blowup may occur aft…
In a recent article, J.-Y. Chemin, I. Gallagher and M. Paicu obtained a class of large initial data generating a global smooth solution to the three dimensional, incompressible Navier-Stokes equations. This data varies slowly in the vertical direction (is a function on $εx_3$) and has a norm which blows up as the small parameter goes to zero. This type of initial data can be seen as the "ill prepared" case (in opposite with the "well prepared" case which was treated previously by J.-Y. Chemin and I. Gallagher). In that paper, the fluid evolves in a special domain, namely $Ω=T^2_h\times\R_v$. T…
This paper deals with the fully parabolic chemotaxis system of local sensing in higher dimensions. Despite the striking similarity between this system and the Keller--Segel system, we prove the absence of finite-time blow-up phenomenon in this system even in the supercritical case. It means that for any regular initial data, independently of the magnitude of mass, the classical solution exists globally in time in the higher dimensional setting. Moreover, for the exponential decaying motility case, it is established that solutions may blow up at infinite time for any magnitude of mass. In order…
We study solutions of the focusing energy-critical nonlinear heat equation $u_t = Δu - |u|^2u$ in $\mathbb{R}^4.$ We show that solutions emanating from initial data with energy and $\dot{H}^1-$norm below those of the stationary solution $W$ are global and decay to zero, via the "concentration-compactness plus rigidity" strategy of Kenig-Merle. First, such global solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza, Seregin and Sverak in an argu…
This paper is concerned with the structure of the solutions to subcritical elliptic equations related to the Matukuma equation. In certain cases the complete structure of the solution set is known, and is comparable to that of the original Matukuma equation. Here we derive sufficient conditions for a more complicated solution set consisting of; (i) crossing solutions for small initial conditions and large initial conditions; (ii) at least one open interval of slowly decaying solutions; and (iii) at least two rapidly decaying solutions. As a consequence we obtain multiplicity results for rapidl…
A Weakly Turbulent solution to the cubic Nonlinear Harmonic Oscillator on $\mathbb{R}^2$ perturbed by a real smooth potential decaying to zero at infinity
We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and also sharp evaluation for the confinement of mass, i.e., the effective speed of propagation. We provide estimates for some moments of the solution, defined using the distance from a given vertex. Our technique relies on suitable inequalities of Faber-Krahn type, and looks at the local theory of continuous nonlinear partial differential equations. As it is k…
We consider the semilinear heat equation in large dimension $d\geq 11$ $$ \partial_t u =Δu+|u| ^{p-1}u, \ \ p=2q+1, \ \ q\in \mathbb N $$ on a smooth bounded domain $Ω\subset \mathbb R^d$ with Dirichlet boundary condition. In the supercritical range $p\geq p(d)>1+\frac{4}{d-2}$ we prove the existence of a countable family $(u_\ell)_{\ell \in \mathbb N}$ of solutions blowing-up at time $T>0$ with type II blow up: $$ \parallel u_{\ell}(t) \parallel_{L^{\infty}} \sim C (T-t)^{-c_\ell} $$ with blow-up speed $c_\ell>\frac{1}{p-1}$. They concentrate the ground state $Q$ being the only radially and d…
