What is the difference between the terms "classical solutions" and "smooth solutions" in the PDE theory? What is the difference between the terms "classical solutions" and "smooth solutions" in the PDE theory? Especially，the difference for the evolution equations? If a solution is in $C^k(0,T;H^m(\Omega))$，can I call it smooth solution?
 A: A classical solution is a function that solves the PDE in the usual sense, ie. $x'=x, x(0)=1 \implies x(t)=e^t$. You can also have weak solutions, which is a variant of the equation with integrals, and is equivalent to the original equation if the solution you are looking at is a classical solution. You can also have solutions as distributions. Look up weak and distribution solutions of Laplace's equation as an example. A smooth solution is one with infinitely many derivatives.
A smooth solution is classical, but a classical solution may not be smooth.
A: A smooth solution is infinitely differentiable. A classical solution is a solution which is differentiable as many times as needed if you want to plug the function into the PDE (for example, if the PDE contains the term $u_{xxxx}$, then the fourth derivate $u_{xxxx}$ must exist in order for $u$ to be a classical solution).
In particular, every smooth solution is a solution in the classical sense. But for the unidirectional wave equation $u_x + u_t = 0$, any function of the form $u(x,t)=f(x-t)$ where $f$ is only (say) twice differentiable, is a classical solution which is not smooth.
