= Positive-time smoothing for a semilinear heat equation
For a <semilinear partial differential equation> $u_t=\Delta u+F(u)$ with smooth $F$ and smooth fixed boundary data, local boundedness permits positive-time <elliptic regularity> and parabolic smoothing. If $u_t$ is continuous with values in a <Hölder space>, elliptic estimates first give continuous $C^{2,\beta}$ spatial regularity for $u$. Time difference quotients then solve a linear parabolic equation with coefficient converging to $F'(u)$; smoothing away from the initial time gives $u_t\in C(C^2)$, hence $u\in C^1(C^2)$. It is essential to gain spatial regularity for the time derivative too; $u\in C(C^2)\cap C^1(C^0)$ alone does not imply $C^1(C^2)$.
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