Journal of the
Korean Mathematical Society JKMS

We show that if $U(x,t)$ is a solution of the heat equation satisfying $$ \int|\partial^\alpha U(x,t)|^pdx0,\quad p>1 $$ then its initial value $U(x,0^+)$ belongs to $W^{p,s}$ which shows the regularity of the initial state. As a corollary, the integral representation of the solutions of the heat equation is given. But this will be seen not true for the case where $p=1$.

Keywords: Sobolev space, heat equation

MSC numbers: 46E35, 35K05