Questions tagged [wave-equation]

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6 votes
1 answer
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 $u$ ...
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0 votes
0 answers
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
1 answer
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
1 answer
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 ...
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3 votes
1 answer
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 ...
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1 vote
1 answer
61 views

Estimate $\Vert \Delta u(t)\Vert_{2}$ in term of energy

We consider the wave equation $$\left\{ \begin{array}{ll} u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\ u=0, \quad u \in \partial \Omega, t>0 \\ u(0,x)=u_{0}(x), \quad u_{t}(0,x)=u_{1}(x), x \...
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2 votes
0 answers
60 views

How to learn Strichartz estimates for wave equations?

I am a student planning to learn some knowledge about Strichartz estimates for wave equations. My goal is to understand the Strichartz estimates or a priori estiamtes of weak solutions for the linear ...
1 vote
0 answers
36 views

Time evolution of Wigner transform

I am studying the Hartree equation for N-particles for the first time and things are not clear to me. Given the density matrix $$\gamma_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) = \begin{cases} \int \...
2 votes
1 answer
171 views

Determine the form of the wave equation in Minkowski space on the line element $ds^2 = -dt^2 + a(t)(dx^2 + dy^2 + dz^2)$

The wave equation in Minkowski space can be given as $-\frac{\partial^2\phi}{\partial t^2}+\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}= ...
2 votes
1 answer
58 views

How to estimate higher order regularity for wave type equation with time dependant coefficients?

Consider the following wave-type equation, $$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$ where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
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4 votes
1 answer
158 views

Energy estimates for nonlinear wave type equation

Consider the following wave-type equation, $$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$ where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
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0 votes
0 answers
54 views

Regularity of solution of wave eq. from regularity of Laplace eq. by Laplace transform

Let us consider the wave equation $$\begin{cases} w_{tt} -\Delta u = 0, & x \in \Omega, \ t >0, \\ w(0,x) = 0, & x \in \Omega,\\ w_t(0.x) = \phi(x), & x \in \Omega,\\ w(t,x) = 0, &...
2 votes
4 answers
220 views

EM-wave equation in matter from Lagrangian

Note I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success. Setup Let's suppose a ...
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0 votes
0 answers
23 views

Does the self inductance and velocity factor of the elements of a dipole antenna change its resonant frequency?

I've asked this question on Ham Stack Exchange but no one knows the answer so i thought would try posting on this site. Below are well known equations from Wikipedia for the resistance and reactance ...
  • 101
6 votes
1 answer
146 views

Fractional derivative notation in wave turbulence

This is my first question in MathOverflow and I will do my best to format it correctly and make it clear. I am reading a paper on dispersive wave turbulence which introduces the following family of ...
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1 vote
1 answer
80 views

Classification of homogeneous distributions

On page 92 of these notes, there is a discussion on how to find the fundamental solution to the D'Alembertian operator. It is firstly proposed that $c_n(t^2 - |x|^2 )^{-(n-1)/2}$ may be a good ...
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3 votes
0 answers
31 views

Partial hypoellipticity

The question is posed specifically about the wave equation and is related to partial hypoellipticity of the wave equation. Let $\Omega \subset \mathbb R^n$, $M=(0,T)\times \Omega$ and $\Gamma=(0,T)\...
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2 votes
0 answers
68 views

Wave equation with infinite time

Let $\Omega \subset \mathbb R^n$ be a compact domain with smooth boundary. Let $f \in H^1_\delta((0,\infty)\times \partial \Omega)$, where $$ \|f\|_{H^1_\delta}^2= \|f\|^2_{L^2_\delta}+\|Df\|^2_{L^2_\...
  • 3,109
6 votes
2 answers
499 views

Non-linear hyperbolic PDE

I have the following PDE in two dimensions $$ 2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0, $$ with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively $$ 2\...
2 votes
0 answers
43 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} \...
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2 votes
0 answers
64 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\,\...
  • 3,109
1 vote
0 answers
52 views

wave equation with $H^{-1}$ source

Let $\Omega$ be a bounded domain with smooth boundary and given $f \in H^{-1}((0,T)\times \Omega)$, consider the wave equation $$ \Box u =f\quad \text{on $(0,T)\times \Omega$},$$ with $u|_{(0,T)\times ...
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2 votes
0 answers
109 views

Support of a fundamental solution of wave equation

The solution of the wave equation $$ \Box E = \delta $$ is $$ E(t,x) = \mathscr{F}^{-1} \left( \frac{\sin (t\lvert \cdot \rvert ) }{\lvert \cdot \rvert} \theta (t) \right)(x)\in\mathcal{S'}(\mathbb{R^{...
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4 votes
1 answer
244 views

Decay estimate on wave equation

In this paper here it is claimed in (1.3) that it is classical and immediate from the explicit solution of the wave equation in 3D $$(\partial_t^2 -\Delta )u(t,x)=0$$ with $u(0,x)=0$ and $u_t(0,x)=g(x)...
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3 votes
1 answer
106 views

wave equation with vanishing trace at infinity

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem \begin{equation}\label{pf0} \begin{aligned} \begin{cases} \Box u+qu=0\,\quad &\text{on $(0,\infty)\times \...
  • 3,109
1 vote
1 answer
225 views

the curvature wave equation

I was referred here from this question I asked on stackexchange. And now that I'm here, I see that this other question about geometric wave equations is very closely related to mine. But I have a ...
0 votes
0 answers
26 views

Approximation of "endpoint" initial data for the 3D wave equation

Consider $f\in L^2(\mathbb{R}^3)$ such that, denoting by $A_f$ the solution to the wave equation $\square A=0$ with initial data $A(0)=0$, $(\partial_t A)(0)=f$, we have $A\in L_2(\mathbb{R}^+;L^{\...
1 vote
0 answers
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)-\...
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2 votes
1 answer
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 ...
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2 votes
0 answers
112 views

About solutions of Klein-Gordon equation

I wonder how to solve the Klein-Gordon equation $$\left\{\begin{split}&\partial_t^2u-\Delta u+u=f\\&u(0,x)=u_1(x),\quad \partial_t u(0,x)=u_2(x)\end{split}\right.$$ where $u(t,x)$ defined on $\...
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2 votes
0 answers
38 views

Some questions about the contruction of center stable manifolds for cubic NLKG by Lyapunov-Perron method

In Nakanish & Schlag: Invariant manifolds and Dispersive Hamiltonian Evolution Equations,, on theorem 3.22, they use Lyapunov-Perron methods to conctruct center stable manifolds for focusing cubic ...
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3 votes
1 answer
118 views

How to find the conserved quantities of the Kirchhoff equation?

Consider the Kirchhoff equation, given by $$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$ where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. ...
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1 vote
3 answers
850 views

Uniqueness of solution of the wave equation

Consider the wave equation $$\frac{\partial^2 u}{\partial t^2}-\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}=0$$ with initial conditions $$u|_{t=0}=\frac{\partial u}{\partial t}|_{t=0}=0$$ Does ...
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2 votes
0 answers
101 views

Inhomogeneous wave equation - a reference

Consider the inhomogeneous wave equation $$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$ where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\...
  • 19.3k
1 vote
0 answers
81 views

Snoidal wave solutions of the $\phi^4$ model

I want to prove the existence of snoidal wave solutions of the $\phi^4$ model, given by $$u_{tt}-u_{xx}=u-u^3,\; (x,t) \in \mathbb{R}\times \mathbb{R}.$$ So, we are looking for solutions in the form $...
  • 205
2 votes
0 answers
62 views

wave equation with non-smooth coefficients

Let us consider the equation $$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$ subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
  • 3,109
1 vote
1 answer
119 views

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^...
  • 409
2 votes
0 answers
57 views

References for numerical approach of Hilbert uniqueness method (HUM)

Finding of the control that achieves the exact controllability of the wave equation (Neumann boundary conditions) using the HUM method (see: J.L. Lions, Controlabilité exacte perturbations et ...
1 vote
0 answers
112 views

Uniform boundedness for Strichartz constants in Tao's book

This is a question from Tao's book "Nonlinear Dispersive Equations". We define the space $S^0(I\times R^d)$ as the closure of the Schwartz functions under the norm $$\|u\|_{S^0}=\sup_{(q,r)\text{ ...
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0 votes
0 answers
127 views

Derivative of a convolution integral of the following type?

I'm looking to find the derivative of a convolution integral of the following form: \begin{equation} \frac{d}{dr}((G(r,t)*f(t)) = \frac{d}{dr} (\int_{-\infty}^{\infty} G(r,t-\tau)f(\tau) d\tau) \end{...
1 vote
0 answers
70 views

Energy inequality - wave equation

In J. L. - Lions book Quelques méthodes de résolution des problèmes aux limites non linéaires, the author proves the following lemma: Lemme 6.1: Let w be a function satisfying $w \in L^\infty(0,...
2 votes
0 answers
54 views

Wave equation with data on null surfaces

Consider the solid cone $C$ as the region inside $z=1-\sqrt{x^2+y^2}$ and bounded by $0\leq z\leq 1$. Now let us define $$\Omega= \{z=\frac{1}{2}\} \cap C\quad \text{and}\quad \Sigma= (\partial C \cap ...
  • 3,109
1 vote
1 answer
73 views

The time when a quasi-linear hyperbolic system produces shocks

I am interested in the time when a quasi-linear $p$-system produces shocks. Let $\mathbb T$ be 1-d torus: $[0, 1]$ with periodic boundary conditions. Fix $p$, $r \in C^\infty(\mathbb T)$. For each ...
2 votes
0 answers
131 views

Strichartz estimates

In the Blair, Smith and Sogge's paper Strichartz estimates for the wave equation on manifolds with boundary, the authors study integrability estimates for solution of the following problem: \begin{...
1 vote
0 answers
24 views

Stabilization of non-autonomuous 1-d wavs equation

I want to ask two questions about the stabilization of the equation $$\eqalign{ & {y_{tt}} = k(t,x){y_{xx}}+a(t,x){y_t}+ b(t,x){y_x}+ c(t,x){y_x} +d(t,x)y \ \ (t,x) \in {\text{ }}(0,\infty ) ...
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1 vote
0 answers
58 views

Wellposedness of semilinear wave equation with discontinuous source

Where can I find existence and uniqueness results for semilinear wave equations with discontinuous, i.e. $$\partial^2_{tt} u - \Delta u = f(u), \quad t >0, \ x \in \Omega$$ where $f$ is ...
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1 vote
0 answers
43 views

Initial-boundary value problem for the damped wave equation with nonlinear source (in bounded domains)

It is well known that the initial-value problem for the wave equation on $\mathbb R^N$ can be studied by means of Fourier transform. What reference presents well-posedness results and qualitative ...
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5 votes
0 answers
142 views

Wave equation with porous medium term

The classical porous media equation is $$u_t - \Delta(u^m) = 0 \quad m>1.$$ Has the (degenerate) wave equation $$u_{tt} - \Delta(u^m) = 0$$ been subject of studies? What would the physical ...
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0 votes
0 answers
69 views

Is the exact solution of the wave equation for the scattering of waves by a disk/cylinder an open problem?

The solution exact solution of the Helmholtz equation for the scattering of waves by a sphere is relatively straightforward and has been known since the time of Lord Rayleigh. The exact solution of ...
2 votes
1 answer
136 views

Well-posedness of wave equations with time-dependent coefficient

Let us consider the following wave equation \begin{array}{rrr} y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in} & (0,T)\times (0,1), \\ y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\ y(0,x)...
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