# Questions tagged [wave-equation]

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### 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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### 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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### 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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### 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 ...
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### 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: \label{pf0} \begin{aligned} \begin{cases} t\partial_t(t\partial_t u)-\...
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### 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^{\...
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### Derivative of a convolution integral of the following type?

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