I am looking for estimates of the Hölder norm for the time-variable of the solution of the heat equation. Let me formulate the problem:
Consider the problem
$\partial_t u(t,x) = \partial_x^2 u(t,x)$, $(t,x) \in [0,T]\times \mathbb{R}$
with initial condition
$u(0,x)=u^0(x)$, $x\in\mathbb{R}$,
where $u^0\in C^{1,\alpha} (\mathbb{R},\mathbb{R})$, i.e. $u^0$ is continuously differentiable and the derivative is (locally) $\alpha$-Hölder continuous ($\alpha$-Hölder continuous on $[-a,a]$, with a Hölder constant depending on $a\in \mathbb{R}$).
Then, the solution is given by $P_t u^0(x)$, where $(P_t)_{t\in [0,T]}$ denotes the heat semigroup.
I want an estimate for
$\left |P_t u^0(x) - u^0(x)\right |$, $x\in [-a,a]$,
in terms of the $\alpha$-Hölder norm of the derivative of $u^0$ on $[-a,a]$.
Do you know such a result or a similar result, or do you have an idea where I could find it?
Thank you for your help! Luke