相关论文: Inward/outward Energy Theory of Wave Equation in H…
We prove decay with respect to some Lebesgue norms for a class of Schr\"odinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space…
We consider the focusing energy-critical nonlinear Schr\"odinger equation $iu_t+\Delta u = - |u|^{\frac4{d-2}}u$ in dimensions $d\geq 5$. We prove that if a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys $\sup_{t\in I}\|\nabla…
In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial_t^2u-\Delta u + \partial_tu = f(u) & \text{in}\ \Omega\times (0,T), \\ u=0 & \text{on}\…
In this work, we consider the energy-supercritical defocusing cubic nonlinear wave equation in dimension d=5 for radially symmetric initial data. We prove that an a priori bound in the critical space implies global well-posedness and…
Under appropriate assumptions the energy of wave equations with damping and variable coefficients $c(x)u_{tt}-\hbox{div}(b(x)\nabla u)+a(x)u_t =h(x)$ has been shown to decay. Determining the rate of decay for the higher order energies…
We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb{R}$. We show an upper bound for any blow-up…
We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation $-u_{tt} + u_{xx} = |u|^{p-1} u$, where $p > 1$. Standard energy methods guarantee global existence, but do not…
In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the…
We consider the total energy decay together with L^2-bound of the solution itself of the Cauchy problem for wave equations with a localized damping and a short-range potential. We treat it in the one dimensional Euclidean space R. We adopt…
In this paper, we prove the decay and scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay…
In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schr\"odinger equation with harmonic potential, $(i\partial_t + \frac {\Delta}2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free…
We consider the energy-supercritical nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ with defocusing cubic nonlinearity in dimension $d=5$ with no radial assumption on the initial data. We prove that a uniform-in-time {\it a priori}…
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…
By definition, the exterior asymptotic energy of a solution to a wave equation on $\mathbb{R}^{1+N}$ is the sum of the limits as $t\to \pm\infty$ of the energy in the the exterior $\{|x|>|t|\}$ of the wave cone. In our previous work (JEMS…
In any dimension $n \geq 3$, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schr\"odinger equation $i u_t + \Delta u = |u|^{\frac{4}{n-2}} u$ in $\R \times \R^n$ exist globally and…
In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= |u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p\in (\frac4d,\frac4{d-2})$. We prove that…
We consider a class of biharmonic nonlinear Schr\"odinger equations with a focusing inhomogeneous power-type nonlinearity \[ i\partial_t u -\Delta^2 u+\mu\Delta u +|x|^{-b} |u|^\alpha u=0, \quad \left. u\right|_{t=0}=u_0 \in…
In this paper we consider the semi-linear wave equation: $u_{tt}-\Delta u=u_t|u_t|^{p-1}$ in $\mathbb{R}^N$ where $1<p\leq 1+\frac2{N-1}$ and $p<5$ if N=1, $p\neq 3$ if N=2. We give an energetic criteria and optimal lower bound for blowing…
Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give…
We study the large time behavior of solutions to the semilinear wave equation with space-dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the…