 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 $F^{\prime }(u)$; 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 $C^1(C^2)$.