# Questions tagged [wave-equation]

The tag has no usage guidance.

76 questions
Filter by
Sorted by
Tagged with
139 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 $u$ ...
• 61
114 views

### Nonlinear wave equation : ODE blow-up construction

Consider the focusing non-linear wave equation (NLW) \begin{align*}\partial_{tt} u(t,x) - \Delta u(t,x) &= -|u|^{p-1}u, \quad (t,x) \in [0,+\infty) \times \mathbb R^d \\ u(0,x) &= u_0(x) \in \...
• 101
1 vote
79 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
78 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 ...
• 101
247 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 ...
• 133
1 vote
61 views

171 views

• 485
158 views

• 41
106 views

• 913
1 vote
64 views

### Well-posedness for a wave equation with degenerate coefficients

Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem: \begin{equation}\label{pf0} \begin{aligned} \begin{cases} t\partial_t(t\partial_t u)-\...
• 3,109
38 views

### What is the analytical form of the cylindrical wave appearing on reflection of a plane wave from a corner?

This is a cross-post from Math.SE, where no answer was given after 3 months. Consider a plane 2D wavelet moving towards a corner reflector with 120° opening angle with infinitely extended sides. The ...
• 285
112 views

• 19.3k
1 vote
81 views

• 3,109
1 vote
119 views

• 653