The wave-equation tag has no usage guidance.

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### Wave kernel for the circle $S^1$ - Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle?
The definitions are related to the page $25$ of the following pdf.
As the ...

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### Fundamental solution on the interval $[0,l]$ - Wave equation [closed]

I found a solution by separation of variables for the wave equation $$\frac{d^2u}{dt^2}-c^2\frac{d^2u}{dx^2}=0,$$ with Dirichlet boundary condition $u(0,t)=u(l,t)=0$ : $$u_n(x,t)=T_n(t)X_n(x)=(k_1 ...

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### Wave equation on the interval $[0,l]$ and wave kernel [closed]

I found a solution by separation of variables for the wave equation $$\frac{d^2u}{dt^2}-c^2\frac{d^2u}{dx^2}=0,$$ with Dirichlet boundary condition $u(0,t)=u(l,t)=0$ : $$u_n(x,t)=T_n(t)X_n(x)=(k_1 ...

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### 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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### Parametrices for the wave equation on manifolds with boundary

I am trying to understand parametrices for the solution operator $G_t = \sin(t\Delta)/\Delta$ to the wave equation
$$(\partial_{tt} + \Delta)u_t=0, ~~~~~~~ u_0 =0, ~~~~~~\partial_tu_0 = f$$
on a ...

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### 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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### 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.

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### Generalized wave equation

I asked this question here:
http://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 ...

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98 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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### 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 ...

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### 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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### 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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### 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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### 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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### 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), ...

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### 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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135 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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### 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 ...