A) It is possible for a wave equation to have as a solution a finite-duration function? Any closed-form example? (please share the specific wave equation with its finite-duration solution, showing how it is a solution – I want to know also How to work with a compact-supported function in more than one dimension).
B) I am specially interested in the classic electromagnetic wave equation $\nabla \vec{E}=\frac{1}{c^2}\frac{\partial^2}{\partial t^2} \vec{E}$, Could it admit compacted-supported solutions?
C) If the classic electromagnetic wave equation can´t sustained finite-duration solutions, Are there any non-linear versions that have compact-supported solutions?
I am specially interested in figure out if finite-duration functions that starts and or ends at a value different from zero could be a solution or not (that is why I am asking for a general finite-duration function). If not possible, also to know why It can´t, and what restrictions have to fulfill a finite-duration function to be an answer to a wave equation. Thinking in a laser pointer, I believe is reasonable to think that the solution function could have at least an ending point different to zero that jumps to zero, since they abruptly goes off, but I don´t know if it could be modeled by the wave equation.
I already know that there exist non-linear versions where Soliton Waves happen, which are highly localized waves, but the function that describes them is vanishing-at-infinity and not a proper finite-duration/compact-supported function (I believe Solitons waves are proportional to the square of a hyperbolic secant function).
Beforehand thanks you very much.
PS: compact-supported means here that there exists and starting time $t_0$ and a ending time $t_F$ such that the function is $f(t) = 0, \forall t<t_0$ and $f(t) = 0, \forall t>t_F$, so is of finite duration. If $f(t)$ is continuous and compact-supported, then also is bounded $\|f(t)\|_\infty < \infty$.
Best Answer
You are confusing compactly supported and finite duration - these do not mean the same thing in the context of PDEs that distinguish between time and spatial variables. Not many people would reasonably assume compactly supported in such a context would refer to the temporal variable. As discussed in the comments a globally finite duration solution violates existence and uniqueness. However, consider the following function $f:\Bbb{R}^3\to\Bbb{R}$
$$f(x,y,z) = \begin{cases}\exp\left[\frac{-1}{R^2-x^2-y^2-z^2}\right] & x^2+y^2+z^2 < R^2 \\ 0 & x^2+y^2+z^2 \geq R^2\end{cases}$$
Then for $k\in\Bbb{R}^3$ with $|k|=1$, we have that
$$E_i(x,y,z,t) = f(k_xx-ct,k_yy-ct, k_zz-ct)$$
satisfies the wave equation and in particular is compactly supported spatially for all times (this is a bubble of radius $R$ traveling in the $k$ direction). Below is an animation of the equivalent expression in 2D instead of 3D travelling in the $45^\circ$ direction