Questions tagged [wave-equation]

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5
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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 ...
5
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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 $$(\partial_{tt} + \Delta)u=0, ~~~~~~~ u_0 =0, ~~~~~~\partial_tu_0 = f$$...
3
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28 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)\...
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58 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_\...
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
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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\,\...
2
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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^{...
2
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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
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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 ...
2
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84 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, $\...
2
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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((...
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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 ...
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54 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
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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{...
2
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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
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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}(...
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50 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|...
2
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399 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^...
2
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62 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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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 ...
1
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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)-\...
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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 $...
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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{ ...
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59 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,...
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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 ) ...
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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 ...
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40 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 ...
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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 ...
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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}+ ...
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0answers
318 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 ...
1
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0answers
110 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} \ ...
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177 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: $\|...
0
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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, &...
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19 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 ...
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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}\...
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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^{\...
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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{...
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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 ...
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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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0answers
34 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.