# Questions tagged [wave-equation]

The wave-equation tag has no usage guidance.

46
questions

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164 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

**0**answers

58 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**

votes

**0**answers

61 views

### how to show that the operator $\mathcal {L}$ has only one negative eigenvalue?

Consider the operator $\mathcal {L}: H^2_{per}([0,L])\subset L^2_{per}([0,L]) \longrightarrow L^2_{per}([0,L])$ given by
$$\mathcal{L}(y)=w\cdot y''+(3\varphi-1)y, \; \forall \; H^2_{per}([0,L]),$$
...

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vote

**0**answers

66 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 $...

**1**

vote

**0**answers

42 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

**1**answer

70 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

**0**answers

47 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

94 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

**0**answers

39 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

49 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

51 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 ...

**1**

vote

**1**answer

61 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

111 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{...

**0**

votes

**0**answers

72 views

### Strichartz estimates for the inhomogeneous wave equation

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

17 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

**0**answers

50 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

**0**answers

31 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 ...

**5**

votes

**0**answers

119 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 ...

**0**

votes

**0**answers

47 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

119 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

**0**answers

73 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

**0**answers

56 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}(...

**1**

vote

**1**answer

111 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

**0**answers

49 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**

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**0**answers

78 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}+ ...

**1**

vote

**0**answers

219 views

### Finite speed of propagation for $u_{tt} - \Delta (u^p) = 0$

Is anything known about the finite speed of propagation of wave-like nonlinear PDE:
$$u_{tt} - \Delta \left(u^p\right) = 0$$
when say $p > 1$?
That is given initial data $u(x,0) = w_1(x)$ and ...

**2**

votes

**1**answer

89 views

### Maxwell-Klein-Gordon energy estimates in Klainerman and Machedon's 1994 paper

In the 1994 paper On the Maxwell-Klein-Gordon Equation with Finite Energy of Klainerman and Machedon, the proof of Proposition 1.1 contains the following statement. For $\phi$ (the scalar field of) a ...

**4**

votes

**1**answer

510 views

### Finite speed propagation by finite energy method

I am currently investigating some finite speed propagation property for nonlinear wave equations, and I am asking myself if there is a way of proving such a property just using energy estimates like ...

**2**

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**0**answers

49 views

### From Boundary to righthandside

I have a problem coming from linear elasticity in $(x,y,z)\in\mathbb{R}^2\times \mathbb{R}^+$, $t\in \mathbb R$:
$$\left\{\begin{aligned}\partial_{tt} \sigma&=A(D_x,D_y,D_z) \sigma\\
\sigma\big|...

**3**

votes

**1**answer

146 views

### Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient?

I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation ...

**1**

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**0**answers

104 views

### $L^\infty(\Omega)$-regularity for strongly damped wave equation

I am interested in the following IBVP for the strongly damped wave equation:
\begin{equation}
u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\
u=0 \quad \text{on} \ ...

**0**

votes

**1**answer

96 views

### Reference request for the focussing example

I was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the American Mathematical Society Vol. ...

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**0**answers

189 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
$$(\partial_{tt} + \Delta)u=0, ~~~~~~~ u_0 =0, ~~~~~~\partial_tu_0 = f$$...

**0**

votes

**0**answers

48 views

### Non interacting complex unit

How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...

**0**

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**0**answers

33 views

### Any reference in absorbing boundary conditions for non-abelian gauge fields?

Is there any paper on absorbing boundary conditions for non-abelian gauge fields?
Currently I only saw some on elastic wave equations and some on EM fields.

**1**

vote

**1**answer

312 views

### Generalized wave equation

I asked this question here:
https://math.stackexchange.com/questions/1160134/generalized-wave-equation
but did not get any response. I hope it is more suitable on mathoverflow.
I am interested in ...

**5**

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**1**answer

151 views

### Blow-Up for Semi-Linear Wave Equations

I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...

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**0**answers

374 views

### Strichartz estimates for the wave equation

Strichartz estimates for the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$ can be stated as
$$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t u\Vert_{C^...

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votes

**1**answer

367 views

### Boundary energy estimate of wave equations

Let $D$ be the unit disk in $\mathbb{R}^{n}$, we consider the $n$ dimension wave equation defined on $D$,
$$\square u=F$$
where $\square=\partial_{t}^{2}-\triangle$ is the standard wave operator in $\...

**2**

votes

**0**answers

61 views

### only solution to wave equation under certain restriction?

Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two bounded solutions to the wave equation
\begin{equation}
\partial_t^2u_i = \partial_x^2u_i
\end{equation}
obeying the restriction
$\...

**1**

vote

**1**answer

118 views

### Wave equation with linear coefficients

The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed ...

**4**

votes

**1**answer

393 views

### a road from virial identity to Strichartz estimates for wave/ Schrodinger eqs?

Is there any way how virial identity implies Strichartz estimates ( or some smoothing properties) for solutions to a) wave equation b) Schrodinger equation ( say in 3d)? To keep things clear I am ...

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votes

**1**answer

195 views

### Besov Characterization of Strichartz Estimate.

On page 4 of this paper of Ibrahim, Majdoub, and Masmoudi, the authors claim in Proposition 2 that solutions to
$\left\{\begin{array}{ll}\square u=F(t,x)\\ u(0,x)=f(x), \partial_tu(0,x)=g(x)\end{...

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vote

**1**answer

199 views

### Double series solution of wave equation

Let $u(x,y,t)$ be the solution of wave equation $u_{tt}=u_{xx}+u_{yy}, 0\lt x\lt 1, 0\lt y\lt 1, t\ge 0,$ $u(x,y,0)=(x-x^2)(y-y^2), u_t(x,y,0)=0$ and $ u(x,y,t)=0$ on the boundary of the square. Then
...

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**0**answers

171 views

### Strichartz estimates over cones

I'm trying to understand Sogge's book Lectures on Non-Linear Wave Equations, the part where he proves global existence for semilinear equations. There is one part he uses the following inequality:
$\|...

**2**

votes

**1**answer

199 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^...