Consider the equation
$
Lu=\sum\limits_{i,j=1}^n a_{ij}(x,t) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum\limits_{i=1}^n b_{i}(x,t) \frac{\partial u}{\partial x_i} + c(x,t)u-\frac{\partial u}{\partial t} = f(x,t) \mathrm{in\ } D\times (0,T]$
$ u(x,0)=\psi(x),\ \mathrm{on\ } \bar{D},$
$
u(x,t)+\frac{\partial u}{\partial \nu} = g(x,t),\ \mathrm{on\ } \partial D \times (0,T].$
Let $u(t,x)$ is a continious function that solves the above equation (in classical sense not weak sense). If the coefficients(i.e, $a_{ij}(x,t),b_{i}(x,t),c(x,t) $ ) are smooth, then we can say that $u(x,t)$ is also smooth ?