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?
PDE Theory – Difference Between Classical and Smooth Solutions
partial differential equationsterminology
Best Answer
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.