[Vibrating String equation]
$$\partial_{tt} u = k^2 \partial_{xx}u$$
Both sides of the equation are meaningful if $u$ is a distribution. If the equality holds, we call $u$ a weak solution. If $u$ is continously differentiable and the equality holds, we call $u$ a classical solution.
page-21, Guide to Distribution Theory and Fourier transforms.
I'm a bit confused on how exactly these two solutions differ. Could someone who is more learned about this topic explain what picture they have in their head when they think of these two categorizations of solutions? Thanks.