Questions tagged [wave-equation]

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2
votes
1answer
125 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
1answer
38 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)...
3
votes
1answer
98 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)...
0
votes
0answers
48 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, &...
5
votes
0answers
134 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 ...
2
votes
4answers
190 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 ...
2
votes
1answer
33 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 ...
0
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0answers
18 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 ...
6
votes
1answer
125 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 ...
1
vote
1answer
59 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 ...
0
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0answers
85 views

uniqueness for a wave equation

Let $\Omega \subset \mathbb R^n$ be a bounded domain with smooth boundary and let $u \in L^2((0,\infty)\times \Omega)$ solve $\Box u=0$ on $(0,\infty)\times \Omega$. Assume, moreover that $\textrm{Tr}\...
3
votes
0answers
27 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)\...
2
votes
0answers
56 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_\...
6
votes
2answers
458 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
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0answers
41 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} \...
2
votes
0answers
62 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
0answers
48 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 ...
2
votes
0answers
96 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^{...
4
votes
1answer
187 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)...
3
votes
1answer
90 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 \...
0
votes
1answer
124 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
0answers
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
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0answers
62 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)-\...
2
votes
0answers
103 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 $\...
2
votes
0answers
32 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 ...
3
votes
1answer
109 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 $r \in \mathbb{N}$. ...
1
vote
3answers
602 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 ...
2
votes
0answers
82 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, $\...
1
vote
0answers
79 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 $...
2
votes
0answers
56 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((...
1
vote
1answer
104 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^...
2
votes
1answer
128 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)...
2
votes
0answers
55 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
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0answers
108 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{ ...
0
votes
0answers
114 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
0answers
58 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
0answers
53 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 ...
2
votes
1answer
201 views

Curvatures of contours of solutions of 3d Poisson's equation

Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial z^...
1
vote
1answer
70 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
0answers
122 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
0answers
21 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 ) ...
1
vote
0answers
57 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 ...
1
vote
0answers
39 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 ...
0
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0answers
63 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
0answers
78 views

Wave equation with 'spring' integral boundary condition

I am really stuck with this small toy problem. I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is: ...
2
votes
0answers
68 views

Exponential decay for wave equation in even dimensions

Consider the wave equation $$ u_{tt} = \Delta_x u - q(x)u, \quad x \in\mathbb R^d, \; t > 0,\tag{1}\\ u(0,x) = u_0(x) \in H^1_\text{comp}(\mathbb R^d),\\ u_t(0,x) = u_1(x) \in L^2_\text{comp}(...
5
votes
0answers
198 views

Parametrices for the wave equation on manifolds with boundary

I am trying to understand parametrices for the solution operator $G_t = \sin(t\sqrt{\Delta})/\sqrt{\Delta}$ to the wave equation $$(\partial_{tt} + \Delta)u=0, ~~~~~~~ u_0 =0, ~~~~~~\partial_tu_0 = f$$...
1
vote
1answer
128 views

Examples of the time-dependent linear wave equation

I am looking for examples of the non-autonomous linear wave equation that have some relevant applications. What articles and sources (maybe, some catalogue like Handbook of Nonlinear Partial ...
1
vote
0answers
52 views

Supnorm problem involving kernel of Cauchy problem

Let $M$ be the $2$-dimensional hyperbolic manifold. Let $K(t,x,y)$ be the kernel appearing in the fundamental solution of the Cauchy problem $$(\partial^2_t-\Delta_M)u=0,\text{ on }\mathbb{R}^+\times ...
1
vote
0answers
81 views

Diffraction across an absorbing wall

I need help finding the procedure for the solution of the following differential equation. This is equation is: Find $u:\mathbb{R}^2 \to \mathbb{C}$ such that for $C>0$ $ \begin{cases} u_{xx}+ ...