Continuity of parabolic equation

Question

$\partial_tf(x,t)=\frac{\partial ^2f}{\partial x^2}$

$f(x,0)=g(x)$

$(x,t)\in\Omega\times(0,T]$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$.

$g(x)$ is smooth function, then I want to know how to show that $f(x,t) $ is continuous about $t,$ and $\Delta f$ is continuous about $t,$ and $f(x,t)$ is smooth about $x$.

Answer 1

Hint: express the solution as a convolution with the heat kernel and use the dominated convergence theorem.