# Questions tagged [wave-equation]

The wave-equation tag has no usage guidance.

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### 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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16 views

### Solution using Fourier transform to IVPs of elastic wave equations

I'm trying to find a solution using the Fourier transform and its inverse to initial value problems of elastic wave equations in two dimensional periodic media.
\begin{equation}
\begin{cases}
\rho u_{...

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70 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:
...

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50 views

### About well-posedness and regularity for a wave equation with nonhomogeneous Dirichlet boundary condition?

I want to get any result about well-posedness and regularity for the following wave equation with nonhomogeneous Dirichlet boundary condition described by
Let $\Omega \subset
%TCIMACRO{\U{211d} }%
%...

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42 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}(...

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**1**answer

87 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 ...

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47 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 ...

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75 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}+ ...

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160 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**

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**1**answer

79 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 ...

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**1**answer

260 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 ...

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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|...

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**1**answer

136 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 ...

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92 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**

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**1**answer

91 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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158 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
$$(\partial_{tt} + \Delta)u=0, ~~~~~~~ u_0 =0, ~~~~~~\partial_tu_0 = f$$...

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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 ...

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31 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

274 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 ...

**4**

votes

**1**answer

138 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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251 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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298 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 $\...

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58 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
$\...

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vote

**1**answer

115 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 ...

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votes

**1**answer

344 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

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172 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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196 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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165 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:
$\|...

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**1**answer

187 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^...