Questions about partial differential equations of hyperbolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
2
votes
0answers
45 views
Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles
I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...
0
votes
0answers
72 views
Wave operator for focusing NLS
Consider the NLS equation
\begin{equation}
\left\{
\begin{array}{rl}
iu_t + \Delta u+u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\right.
\end{equation}
where ...
5
votes
1answer
125 views
On a conjecture of Lions for the wave equation
In Control of Distributed Singular Systems p 236, JL Lions makes the conjecture :
Let $\Omega$ be a domain in $\mathbb{R}^n$, $Q = \Omega \times ]0,T[$ and consider
$\phi'' - \triangle \phi = F$
...
3
votes
0answers
84 views
Rarefaction Shock Wave Interaction
I am interested in explicit solutions in 1D for the interaction of a rarefaction wave with a shock wave of arbitrary strength. The book Supersonic Flow and Shock Waves by Courant and Friedricks ...
0
votes
0answers
62 views
Decay for linear wave equations
According to Klainerman's global Sobolev inequality, we can show the one derivatives of the free wave decays as $\frac{1}{t}$ on Minkowski spacetime $\mathbb{R}^{3+1}$. But how can we show that the ...
1
vote
1answer
137 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
...
3
votes
0answers
203 views
well-posedness of the transport equation
I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
