# Questions tagged [wave-equation]

The tag has no usage guidance.

88 questions
Filter by
Sorted by
Tagged with
80 views

54 views

34 views

### Understanding the boundary condition of spherical waves in the flat spacetime

I am trying to understand one of the two boundary conditions one has to impose to find the solutions of the wave equation in the flat space-time inside a collapsing null shell. For the spherical wave, ...
35 views

### Are standing waves equivalent to travelling waves as a modelling tool to solve wave equation?

I am learning Fourier Analysis from Elias Stein's excellent textbook. He starts off by explaining difference between standing waves and travelling waves and then demonstrating how you can either to ...
1 vote
127 views

### Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?

We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}d\xi$ is a solution to the homogeneous Schrodinger equation with initial ...
272 views

### Deriving Sommerfeld radiation condition from limiting absorption principle

For the Helmholtz equation $$-(\Delta + k ^2) u = f, \label{1}\tag{1}$$ imposing the Sommerfeld radiation condition $$\lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0$$ on $u$ ...
1 vote
102 views

### Wave equation in $\Omega\times(0,T)$

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^d$ and $T>0$ be a positive number. Consider the wave equation in the domain $\Omega\times(0,T)$ \begin{align} \left\{\begin{matrix} \...
1 vote
131 views

### Finite propagation speed for non-smooth solutions to nonlinear wave equation

Consider the semilinear wave equation in $[0,t_0] \times \mathbb R^d$ : $$\square u = \pm |u|^{p-1}u$$ With subcritical/critical power $1 < p \leq \frac{d+2}{d-2}$. It is easy to show by energy ...
277 views

### On a nonlinear wave equation

I am considering the following wave equation (for $\phi=\phi(x,t)$) $$\phi_{tt} - \Delta \phi = a |\nabla \phi|^2, \quad (x,t) \in \mathbb{R}^3 \times \mathbb{R}$$ where $\nabla$ is just spatial ...
1 vote
69 views

232 views

231 views

97 views

45 views

### A question for regularity of solutions to wave equation

let $T>0$ and suppose $\Omega$ is a bounded domain with smooth boundary. Let $g \in L^\infty(0,T;H^1(\partial \Omega))$ and consider the wave equation \begin{equation}\label{pf0} \begin{aligned} \...
69 views

### wave equation with L^2 boundary data via spectral decomposition

It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation \begin{equation}\label{pf2} \begin{aligned} \begin{cases} \partial^2_{t}u- \Delta u=0\,\...
1 vote
59 views

1 vote
### the energy of scattering solution of cubic Klein-Gordon equation in $n=3$?
Let $n=3$ and $u$ be the solution to Klein-Gordon equation \begin{equation} \begin{cases}\ddot{u}-\Delta u +u=u^3 \\ u(0)=u_0, \partial_t u(0)=u_1, \end{cases} \end{equation} where \$(u_0,u_1) \in H^...