J. Korean Math. Soc. 1996; 33(4): 1039-1046
Printed December 1, 1996
Copyright © The Korean Mathematical Society.
Soon-Yeong Chung
Sogang University
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
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