We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R^d)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \mathbb R^d \to \mathbb R_+$ be the distributional solution to the non-linear PDE $$ \partial_t u (t, x) = \Delta_x \sigma (u (t, x)). $$

For brevity, let $u_t := u(t, \cdot)$. We assume that

• $u_0 \in C^{0, \beta}_b (\mathbb R^d)$ for some $\beta \in (0, \alpha)$.
• there is $\lambda >0$ such that $\frac{1}{\lambda} \le \sigma \le \lambda$.

Are there some estimates of $\| u_t \|_{C^{0, \beta}_b}$ in terms of $u_0$ and $\sigma$?

Any reference is greatly appreciated! Thank you so much for your help!