- #1

evinda

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I want to prove that if for the initial value problem of the wave equation

$$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$

the data (i.e. the initial data and the non-homogeneous $f$) have compact support, then, at each time, the solution has compact support.

I have thought the following.

Suppose that we have the initial data $u(x,0)=\phi(x)$ and $u_t(x,0)=\psi(x)$.

The functions $f, \phi, \psi$ have compact support, meaning that the functions are zero outside a bounded set $[a,b]$.

The solution of the initial value problem is given by

$$u(x,t)=\frac{1}{2}[\phi(x+t)+\phi(x-t)]+\frac{1}{2}\int_{x-t}^{x+t} \psi(y) dy+\frac{1}{2} \int_0^t \int_{x-(t-s)}^{x+(t-s)} f(y,s)dy ds$$

Let $t=T$ arbitrary.

Then

$$u(x,T)=\frac{1}{2}[\phi(x+T)+\phi(x-T)]+\frac{1}{2}\int_{x-T}^{x+T} \psi(y) dy+\frac{1}{2} \int_0^T \int_{x-(T-s)}^{x+(T-s)} f(y,s)dy ds$$

We check when $u(x,T)=0$.

We have $u(x,T)=0$ when

- $x+T, x-T \in \mathbb{R} \setminus{[a,b]}$,
- $x-T,x+T<a$ or $x-T,x+T>b$,
- $x-(T-s)<a$ and $x+(T-s)<a$ or $x-(T-s)>b$ and $x+(T-s)>b$,

Thus $u$ is non-zero outside $[a-T,b+T]$ and $u$ has compact support. Is everything right? (Thinking)