Let $u(x,t)=f(x+t)$ , where $f$ is any locally integrable function on $\mathbb{R}$. Show that $u_{tt}=u_{xx}$ in the sense of Distributions
My try:
For $\phi \in D(\mathbb{R})$, $$(u_{tt},\phi)=-(u_t,\frac{d\phi}{dt})=(u,\frac{d^2\phi}{dt^2})=\int_{K}f(x+t)\frac{d^2\phi}{dt^2} dt$$ $$(u_{xx},\phi)=-(u_x,\frac{d\phi}{dx})=(u,\frac{d^2\phi}{dx^2})=\int_{K}f(x+t)\frac{d^2\phi}{dx^2} dx$$ where $K$ is the support of $\phi$.
These two above are the same and hence they agree.
Is this alright??
Thanks for the help!1
The integrals are not correct. Note that $K\subset\mathbb R^2$. So, you have to integrate with respect to $dt\,dx$. Then use Schwartz and Fubini to see that the integrals agree. For Fubini, better integrate over $\mathbb R^2$ or a square containing $K$ instead of over $K$.