arXiv subjects
Publications and source records attributed to Ruipeng Shen.
We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - Δu = -|u|^{p-1} u$ in dimension $d \in \{3,4,5\}$ and prove the exterior scattering of solutions if $3\leq d \leq 5$ and $1+6/d t+R} |\nabla_{x,t} u(x,t)- \nabla_{x,t} u_L(x,t)|^2 dx = 0 \] for any fixed real number $R$. This generalize the previously known exterior scattering result in the radial case.
In this work we consider the defocusing nonlinear wave equation in one-dimensional space. We show that almost all energy is located near the light cone $|x|=|t|$ as time tends to infinity. We also prove that any light cone will eventually contain some energy. As an application we obtain a result about the asymptotic behaviour of solutions to focusing one-dimensional wave equation with compact-supported initial data.
In this work we consider the energy subcritical 3D wave equation $\partial_t^2 u - Δu = \pm |u|^{p-1} u$ and discuss its (weakly) non-radiative solutions, i.e. the solutions defined in an exterior region $\{(x,t): |x|>|t|+R\}$ with $R\geq 0$ satisfying \[ \lim_{t\rightarrow \pm\infty} \int_{|x|>|t|+R} \left(|\nabla u(x,t)|^2 + |u_t(x,t)|^2\right) dx = 0. \] It has been known that any radial weakly non-radiative solution to the linear wave equation is a multiple of $1/|x|$. In addition, any radial weakly non-radiative solutions $u$ to the energy critical wave equation must possess a similar asymptotic behaviour, i.e. $u(x,t)\simeq C/|x|$ when $|x|$ is large. In this work we give examples to show that radial weakly non-radiative solutions to energy subcritical equation ($3<p<5$) may possess a much different asymptotic behaviour. However, a radial weakly non-radiative solution $u$ with initial data in the critical Sobolev space $\dot{H}^{s_p}\times \dot{H}^{s_p-1}(\mathbb{R}^3)$ must coincide with a $C^2$ solution $W$ to the elliptic equation $-ΔW = -|W|^{p-1} W$ so that $u(x,t) \equiv W(x) \simeq C/|x|$ when $|x|$ is large.
We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - Δu = -|u|^{p-1} u$ in 4 to 6 dimensional spaces with radial initial data. We define $w=r^{(d-1)/2} u$, reduce the equation above to one-dimensional equation of $w$ and apply method of characteristic lines. This gives scattering of solutions outside any given light cone as long as the energy is finite. The scattering in the whole space can also be proved if we assume the energy decays at a certain rate as $x\rightarrow +\infty$. This generalize the 3-dimensional results in Shen[27] to higher dimensions.
We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form equations, and the local boundedness turns out to be sharp in more than two dimensions, answering the `Moser gap' problem left open in arXiv:1506.09203v5. Finally we obtain a maximum principle for weak solutions under the same condition on the degeneracy.
We obtain a generalization of the DeGiorgi Lemma to the infinitely degenerate regime and apply it to obtain continuity of weak solutions to certain infinitely degenerate equations. This reproduces the continuity result obtained in arXiv:1506.09203 via Moser iteration, but only for homogeneous equations. However, the proofs are much less technical and more transparent.
We consider the semi-linear, defocusing wave equation $\partial_t^2 u - Δu = -|u|^{p-1} u$ in $\mathbb{R}^d$ with $1+4/(d-1)\leq p < 1+4/(d-2)$. We generalize the inward/outward energy theory and weighted Morawetz estimates in 3D to higher dimensions. As an application we show the scattering of solutions if the energy of initial data decays at a certain rate as $|x| \rightarrow \infty$.
The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - Δu = - |u|^{p -1} u$ in the 3-dimensional space with $3\leq p<5$. We generalize inward/outward energy theory and weighted Morawetz estimates for radial solutions to the non-radial case. As an application we show that if $3 \frac{5-p}{2}$, then the solution scatters as long as the initial data $(u_0,u_1)$ satisfy \[ \int_{{\mathbb R}^3} (|x|^κ+1)\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{p+1}|u_0|^{p+1}\right) dx < +\infty. \] If $p=3$, we can also prove the scattering result if initial data $(u_0,u_1)$ are contained in the critical Sobolev space and satisfy the inequality \[ \int_{{\mathbb R}^3} |x|\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{4}|u_0|^{p+1}\right) dx < +\infty. \] These assumptions on the decay rate of initial data as $|x| \rightarrow \infty$ are weaker than previously known scattering results.
The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - Δu = - |u|^{p -1} u$ in the 3-dimensional space ($3\leq p<5$) whose initial data are radial and come with a finite energy. In this work we prove scattering in the positive time direction by only assuming the inward part of the energy decays at a certain rate, as long as the total energy is finite, regardless of the decay rate or size of the outward energy. More precisely, we assume the initial data comes with a finite energy and \[ \int_{{\mathbb R}^3} \max\{1,|x|^κ\}\ (\ |\nabla u_0(x)\cdot \frac{x}{|x|} + \frac{u_0(x)}{|x|} + u_1(x)\ |^2 + \frac{2}{p+1}|u_0(x)|^{p+1}\ ) dx < \infty. \] Here $κ\geq κ_0(p) = \frac{5-p}{p+1}$ is a constant. If $κ>κ_0(p)$, we can also prove $\|u\|_{L^p L^{2p}}({\mathbb R}^+ \times {\mathbb R}^3)< +\infty$ and give an explicit rate of $u$'s convergence to a free wave.
The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - Δu = - |u|^{p -1} u$ in the 3-dimensional space ($3\leq p<5$) whose initial data are radial and come with a finite energy. We split the energy into inward and outward energies, then apply energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: "scattering energy" which concentrates around the light cone $|x|=|t|$ and moves to infinity at the light speed and "retarded energy" which is at a distance of at least $|t|^β$ behind when $|t|$ is large. Here $β$ is an arbitrary constant smaller than $β_0(p) = \frac{2(p-2)}{p+1}$. A combination of this property with a more detailed version of the classic Morawetz estimate gives a scattering result under a weaker assumption on initial data $(u_0,u_1)$ than previously known results. More precisely, we assume \[ \int_{{\mathbb R}^3} (|x|^κ+1)\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{p+1}|u|^{p+1}\right) dx < +\infty. \] Here $κ>κ_0(p) =1-β_0(p) = \frac{5-p}{p+1}$ is a constant.
In this short paper we consider a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - Δu = - |u|^{p -1} u$ in the 3-dimensional space with $p \in (3,5)$. We prove that if the energy of radial initial data $(u_0, u_1)$ outside a ball of radius $r$ centred at the origin decays faster than a certain rate $r^{-κ(p)}$, then the corresponding solution $u$ must scatter in both two time directions. The main tool of our proof is a more detailed version of the classic Morawetz estimate.
We develop subrepresentation inequalities for infinitely degenerate metrics, and obtain corresponding Poincare and Sobolev inequalities. We then derive conditions on the degenerate metric under which weak solutions to associated infinitely degenerate equations with rough coefficients are locally bounded, satisfy a maximum principle, or are continuous. As an application we obtain W-hypoellipticity of certain infinitely degenerate quasilinear equations with smooth coefficients having mild nonlinearities and degeneracies.
In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - Δu = - |u|^{p -1} u$ in the 3-dimensional space with $p \in [3,5)$. We prove that if initial data $(u_0, u_1)$ are radial so that $\|\nabla u_0\|_{L^2 ({\mathbb R}^3; dμ)}, \|u_1\|_{L^2 ({\mathbb R}^3; dμ)} \leq \infty$, where $d μ= (|x|+1)^{1+2\varepsilon}$ with $\varepsilon > 0$, then the corresponding solution $u$ must exist for all time $t \in {\mathbb R}$ and scatter. The key ingredients of the proof include a transformation $\mathbf{T}$ so that $v = \mathbf{T} u$ solves the equation $v_{ττ} - Δ_y v = - \left(\frac{|y|}{\sinh |y|}\right)^{p-1} e^{-(p-3)τ} |v|^{p-1}v$ with a finite energy, and a couple of global space-time integral estimates regarding a solution $v$ as above.
In this paper we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space ${\mathbb H}^n$ with $3 \leq n \leq 5$: \[ \partial_t^2 u - (Δ_{{\mathbb H}^n} + ρ^2) u = ζ|u|^{4/(n-2)} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}. \] Here $ζ= \pm 1$ and $ρ= (n-1)/2$ are constants. We introduce a family of Strichartz estimates compatible with initial data in the energy space $H^{0,1} \times L^2 ({\mathbb H}^n)$ and then establish a local theory with these initial data. In addition, we prove a Morawetz-type inequality \[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} \frac{ρ(\cosh |x|) |u(x,t)|^{2n/(n-2)}}{\sinh |x|} dμ(x) dt \leq n {\mathcal E}, \] in the defocusing case $ζ= -1$, where ${\mathcal E}$ is the energy. Moreover, if the initial data are also radial, we can prove the scattering of the corresponding solutions by combining the Morawetz-type inequality, the local theory and a pointwise estimate on radial $H^{0,1}({\mathbb H}^n)$ functions.
In this work we consider an energy subcritical semi-linear wave equation ($3 < p < 5$) \[ \partial_t^2 u - Δu = ϕ(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] with initial data $(u,u_t)|_{t=0} = (u_0,u_1)\in \dot{H}^{s_p} \times \dot{H}^{s_p-1}({\mathbb R}^3)$, where $s_p = 3/2 - 2/(p-1)$ and the function $ϕ: {\mathbb R}^3 \rightarrow [-1,1]$ is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation $-ΔW = ϕ(x) |W|^{p-1} W$ has a nonzero radial solution $W \in C^2 ({\mathbb R}^3) \cap \dot{H}^{s_p} ({\mathbb R}^3)$, any radial solution $u$ with a finite uniform upper bound on the critical Sobolev norm $\|(u(\cdot,t), \partial_t u(\cdot,t))\|_{\dot{H}^{s_p}\times \dot{H}^{s_p}({\mathbb R}^3)}$ for all $t$ in the maximal lifespan must be a global solution in time and scatter.
In this work we consider a semi-linear energy critical wave equation in ${\mathbb R}^d$ ($3\leq d \leq 5$) \[ \partial_t^2 u - Δu = \pm ϕ(x) |u|^{4/(d-2)} u, \qquad (x,t)\in {\mathbb R}^d \times {\mathbb R} \] with initial data $(u, \partial_t u)|_{t=0} = (u_0,u_1) \in \dot{H}^1 \times L^2 ({\mathbb R}^d)$. Here the function $ϕ\in C({\mathbb R}^d; (0,1])$ converges to zero as $|x| \rightarrow \infty$. We follow the same compactness-rigidity argument as Kenig and Merle applied on the Cauchy problem of the equation \[ \partial_t^2 u - Δu = |u|^{4/(d-2)} u \] and obtain a similar result when $ϕ$ satisfies some technical conditions. In the defocusing case we prove that the solution scatters for any initial data in the energy space $\dot{H}^1 \times L^2$. While in the focusing case we can determine the global behaviour of the solutions, either scattering or finite-time blow-up, according to their initial data when the energy is smaller than a certain threshold.
In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space \[ \partial_t^2 u - (Δ_{{\mathbb H}^n} + ρ^2) u = - |u|^{p-1} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}; \] and introduce a Morawetz-type inequality \[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} |u|^{p+1} dμdt < C E, \] where $E$ is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in $H^{1/2,1/2} \times H^{1/2,-1/2}({\mathbb H}^n)$ if $2 \leq n \leq 6$ and $1 0. \]
In this paper, we consider the wave equation in 3-dimensional space with an energy-subcritical nonlinearity, either in the focusing or defocusing case. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined in time and scatters. The proof depends on the compactness/rigidity argument, decay estimates for radial, "compact" solutions, gain of regularity arguments and the "channel of energy" method.