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Ruipeng Shen

Publications and source records attributed to Ruipeng Shen.

At least 19 recordsLinked to original sources
arXiv2026

Nonexistence of multi-bubble radial solutions to the 3D energy critical wave equation

In this work we consider the focusing, energy-critical wave equation in 3D radial case. It has been verified that any global or type II blow-up solution decomposes into a superposition of several decoupled grounds states, a free wave and a small error, as time tends to infinity or the blow-up time. This is usually called soliton resolution. However, all known examples of soliton resolution in the 3D radial case come with no more than one soliton. In this work we prove the nonexistence of any global or type II blow up solution with two or more solitons, thus give a complete classification of asymptotic behaviours of radial solutions.

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arXiv2026

Nonexistence of pure bubble type II blow-up solutions to energy critical wave equation in the 3D radial case

In this work we consider the focusing, energy-critical wave equation in 3D radial case. According to the soliton resolution conjecture, which has been verified in the radial case, any type II blow-up solution decomposes into a superposition of several decoupled grounds states, a free wave and a small error, as time tends to the blow-up time. We prove that there does not exist any pure bubble type II blow up solutions. In other words, the free wave part is never zero in the soliton resolution of type II blow-up solutions, regardless of the bubble number.

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arXiv2025

Non-radiative solutions and long-time dynamics of 5D focusing energy-critical wave equation in the radial case

In this article we discuss the long-time dynamics of the radial solutions to the focusing energy-critical wave equation in 5-dimensional space. We give some details about the asymptotic behaviour, topological structure and time evolution of the non-radiative solutions to this equation. As an application we prove a quantitative version of soliton resolution theorem for solutions defined for all time $t>0$, which immediately verifies the soliton resolution conjecture in the radial case, without a priori boundedness assumption on the energy norm of solution as time tends to infinity. The main tool of this work is the radiation theory of wave equations and the major observation of this work is a correspondence between the radiation and the soliton collision behaviour of solutions.

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arXiv2025

Dynamics of 3D focusing, energy-critical wave equation with radial data

In this article we discuss the long-time dynamics of the radial solutions to the energy-critical wave equation in 3-dimensional space. Given a solution defined for all time $t\geq 0$, we show that the soliton resolution phenomenon happens at all times $t>0$ except for a few relatively short time intervals. The main tool is the radiation theory of wave equations and the major observation of this work is a correspondence between the energy radiation and the soliton resolution/collision behaviour of solutions. We also give a few applications of the main observation on the type II blow-up solutions and ``one pass'' theory near pure mutli-solitons.

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arXiv2025

Modified wave operators and scattering for linear wave equations with a repulsive potential

In this work we consider the wave equation with a repulsive potential, either on the half line ${\mathbb R}^+$ or the Euclidean space ${\mathbb R}^d$ with $d\geq 3$. We combine the operator theory and the inward/outward energy theory to deduce a modified wave operator for repulsive potentials decaying like $|x|^{-\beta}$ with $\beta>1/3$. In particular the regular wave operator without modification exists if $\beta>1$. This implies that the asymptotic behaviour of finite-energy solutions to the wave equation $u_{tt} - \Delta u + |x|^{-\beta} u =0$ is similar to that of the solutions to the classic wave equation if $\beta \in (1,2)$.

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arXiv2024

On the wave equation with Coulomb potential

Wave/Schr\"{o}dinger equations with potentials naturally originates from both the quantum physics and the study of nonlinear equations. The distractive Coulomb potential is a quantum mechanical description of distractive Coulomb force between two particles with the same charge. The spectrum of the operator $-\Delta +1/|x|$ is well known and there are also a few results on the Strichartz estimates, local and global well-posedness and scattering result about the nonlinear Schr\"{o}dinger equation with a distractive Coulomb potential. In the contrast, much less is known for the global and asymptotic behaviour of solutions to the corresponding wave equations with a Coulomb potential. In this work we consider the wave equation with a distractive Coulomb potential in dimensions $d\geq 3$. We first describe the asymptotic behaviour of the solutions to the linear homogeneous Coulomb wave equation, especially their energy distribution property and scattering profiles, then show that the radial finite-energy solutions to suitable defocusing Coulomb wave equation are defined for all time and scatter in both two time directions, by establishing a family of radial Strichartz estimates and combining them with the decay of the potential energy.

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arXiv2023

Channel-localized Strichartz estimates of radial wave equations

In this work we give a few new Strichartz estimates of radial solutions to the wave equation. These Strichartz estimates still use $L^p L^q$ type norms in each channel-like region $\{(x,t): |t|+2^k < |x| < |t|+2^{k+1}\}$, with weaker restrictions on $p, q$ than the classic ones, but combine these localized norms together in the way of an $l^2$ space. We also give an application of these Strichartz estimates on the well-posedness theory of non-linear wave equations.

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arXiv2023

Decay Estimates of High Dimensional Adjoint Radon Transforms

In this paper we prove an optimal $L^2-L^{2d}$ decay estimate of the adjoint Radon transform of compactly supported data in $d$-dimensional space via a geometric method. A similar problem in dimension $3$ has be considered in the author's previous work. This work deals with all higher dimensional case $d\geq 4$. As an application we give the decay of Strichartz norms of $5$-dimensional non-radiative free waves. The general idea is similar to the lower dimensional case but we introduce a new method to prove the corresponding geometric inequality because the old method becomes too complicated in higher dimensions.

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arXiv2023

The Moser method and boundedness of solutions to infinitely degenerate elliptic equations

We show that if $\mathbb{R}^{n}$ is equipped with certain non-doubling metric and an Orlicz-Sobolev inequality holds for a special family of Young functions $\Phi $, then weak solutions to quasilinear infinitely degenerate elliptic divergence equations of the form $$\mathrm{div}\mathcal{A}\left( x,u\right) \nabla u=\phi _{0}-\mathrm{div}_{A} \vec{\phi}_{1}$$ are locally bounded. Furthermore, we establish a maximum principle for solutions whenever a global Orlicz-Soblev estimate is available. We obtain these results via the implementation of a Moser iteration method, what constitutes the first instance of such technique applied to infinite degenerate equations. These results partially extend previously known estimates for solutions of these equations but for which the right hand side did not have a drift term. We also obtain bounds for small negative powers of nonnegative solutions; these will be applied to obtain continuity of solutions in a subsequent paper.

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arXiv2022

The radiation theory of radial solutions to 3D energy critical wave equations

In this work we consider a wide range of energy critical wave equation in 3-dimensional space with radial data. We are interested in exterior scattering phenomenon, in which the asymptotic behaviour of a solutions $u$ to the non-linear wave equation is similar to that of a linear free wave $v_L$ in an exterior region $\{x: |x|>R+|t|\}$, i.e. \[ \lim_{t\rightarrow \pm \infty} \int_{|x|>R+|t|} (|\nabla(u-v_L)|^2 + |u_t-\partial_t v_L|^2) dx = 0. \] We classify all such solutions for a given linear free wave $v_L$ in this work. We also give some applications of our theory on the global behaviours of radial solutions to this kind of equations. In particular we show the scattering of all finite-energy radial solutions to the defocusing energy critical wave equations.

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arXiv2022

Classification of radial non-radiative solutions to the 5D nonlinear wave equations

In this work we classify all radial non-radiative solutions to the 5D nonlinear wave equations with a wide range of energy critical nonlinearity. We show that such a solution always comes with two characteristic numbers. These characteristic numbers can be determined by either the radiation profile of the initial data or the asymptotic behaviour of the solution. In addition, two radial weakly non-radiative solutions with the same characteristic numbers must coincide with each other in the overlap part of their exterior regions. Finally we give a few applications of our theory on the global behaviours of solutions to the nonlinear wave equations.

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arXiv2022

An inequality regarding non-radiative linear waves via a geometric method

In this work we consider the operator \[ (\mathbf{T} G) (x)= \int_{\mathbb{S}^2} G(x\cdot ω, ω) dω, \quad x\in \mathbb{R}^3, \; G\in L^2(\mathbb{R}\times \mathbb{S}^2). \] This is the adjoint operator of the Radon transform. We manage to give an optimal $L^6$ decay estimate of $\mathbf{T} G$ near the infinity by a geometric method, if the function $G$ is compactly supported. As an application we give decay estimate of non-radiative solutions to the 3D linear wave equation in the exterior region $\{(x,t)\in \mathbb{R}^3 \times \mathbb{R}: |x|>R+|t|\}$. This kind of decay estimate is useful in the channel of energy method for wave equations

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arXiv2022

Radiation fields and non-radiative solutions to the energy sub-critical wave equations

Radiation field and channel of energy method have become important tools in the study of nonlinear wave equations in recent years. In this work we give basic theory of radiation fields of free waves in the energy sub-critical case. We also show that the asymptotic behaviours of non-radiative solutions to a wide range of non-linear wave equations resemble those of non-radiative free waves. Our theory is completely given in the critical Sobolev spaces of the corresponding nonlinear wave equation and avoids any assumption on the energy of the solutions.

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arXiv2022

Asymptotic behaviour of non-radiative solution to the wave equations

In this work we consider weakly non-radiative solutions to both linear and non-linear wave equations. We first characterize all weakly non-radiative free waves, without the radial assumption. Then in dimension 3 we show that the initial data of non-radiative solutions to a wide range of nonlinear wave equations are similar to those of non-radiative free waves in term of asymptotic behaviour.

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arXiv2021

Energy distribution of solutions to defocusing semi-linear wave equation in higher dimensional space

The topic of this paper is a semi-linear, defocusing wave equation $u_{t t}-Δu=-|u|^{p-1} u$ in sub-conformal case in the higher dimensional space whose initial data are radical and come with a finite energy. We prove some decay estimates of the the solutions if initial data decay at a certain rate as the spatial variable tends to infinity. A combination of this property with a method of characteristic lines give a scattering result if the initial data satisfy $$E_κ\left(u_{0}, u_{1}\right)=\int_{\mathbb{R}^{d}}\left(|x|^κ+1\right)\left(\frac{1}{2}\left|\nabla u_{0}(x)\right|^{2}+\frac{1}{2}\left|u_{1}(x)\right|^{2}+\frac{1}{p+1}\left|u_{0}(x)\right|^{p+1}\right) d x<+\infty.$$ Here $κ=\frac{(2-d)p+(d+2)}{p+1}$.

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arXiv2021

Energy Distribution of solutions to defocusing semi-linear wave equation in two dimensional space

We consider finite-energy solutions to the defocusing nonlinear wave equation in two dimensional space. We prove that almost all energy moves to the infinity at almost the light speed as time tends to infinity. In addition, the inward/outward part of energy gradually vanishes as time tends to positive/negative infinity. These behaviours resemble those of free waves. We also prove some decay estimates of the solutions if the initial data decay at a certain rate as the spatial variable tends to infinity. As an application, we prove a couple of scattering results for solutions whose initial data are in a weighted energy space. Our assumption on decay rate of initial data is weaker than previous known scattering results.

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arXiv2020

Exterior scattering of non-radial solutions to energy subcritical wave equations

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.

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