Definition
Classical solution of a PDE
A function with the required classical derivatives that satisfies a PDE pointwise.
A classical solution of a PDE is a function whose derivatives appearing in the equation exist with the stipulated classical regularity and satisfy the equation pointwise. A smooth solution has continuous derivatives of every order in its open space-time domain.
Example of the required regularity
For a second-order parabolic equation, one common classical class has continuous first time derivatives and continuous spatial derivatives through order two. Such a solution need not be smooth merely by definition; further regularity may follow from the equation and its coefficients.
Initial and boundary values
Regularity on an open time interval does not by itself specify behavior at its endpoints. An initial condition may require continuity to pointwise or in a chosen function-space norm. Boundary conditions and the regularity used to interpret them are additional parts of the problem.