a classical solution

What does it even mean to 'solve' a PDE? The most direct answer — the one you start with — is a classical solution: an honest, ordinary function u that is differentiable enough to have all the derivatives the equation mentions, and that makes the equation literally true at every point when you plug it in. If the equation is u_t = k u_xx, a classical solution is a function whose u_t and u_xx both genuinely exist as ordinary derivatives and satisfy that identity everywhere in the region you care about.

Concretely, 'differentiable enough' means: a second-order equation needs a solution with continuous second derivatives (so the highest derivatives in the equation actually make sense and are continuous), usually written as 'u is C^2'. You then substitute u and its derivatives into the equation and check it holds at each point, plus check that u meets the boundary and initial conditions. There is nothing slippery here — it is the schoolbook notion of a solution, requiring only that the function be smooth enough and obey the rule pointwise. For a verification example: u(x, t) = e^(-k t) sin(x) has u_t = -k e^(-k t) sin(x) and u_xx = -e^(-k t) sin(x), and indeed u_t = k u_xx, so it is a classical solution of the heat equation.

The catch — and it is a big one — is that classical solutions sometimes do not exist, even for sensible-looking problems. Shock waves in fluids develop discontinuities, so no differentiable function can satisfy the equation across the shock; solutions built from a sharp-cornered initial shape may have kinks where second derivatives blow up; data with corners produces solutions with corners. This is precisely why the subject later introduces weak solutions, distributional solutions, and viscosity solutions — broader notions of 'solution' that allow non-smooth functions while still capturing the right physics. A classical solution is the gold standard when you can get one, but much of modern PDE theory exists exactly because you often cannot.