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Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.

We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R^d \to \mathbb R$ 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 linear PDE $$ \partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}. $$

We assume that

  • $u(0, \cdot) \in C^\infty_c (\mathbb R^d)$ is a probability density function.
  • there is $\lambda >0$ such that $\frac{1}{\lambda} \le \sigma \le \lambda$.
  • $\beta \in (0, \alpha)$.

Are there some estimates of $\| u(t, \cdot) \|_{C^{0, \beta}_b}$ in terms of $u(0, \cdot)$ and $\sigma$?

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

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    $\begingroup$ I think the following document might be interesting for you: people.math.harvard.edu/~spicard/notes-parabolicpde.pdf It contains many estimates for a class of parabolic equations which I think includes yours. They use a kind of space-time Hölder space but it should be possible to extract some results from those which are similar to what you are looking for. $\endgroup$
    – Robert Wegner
    Commented Feb 15 at 12:37
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    $\begingroup$ The function $v := \sigma^2u$ solves $\partial_t v = \sigma^2\Delta v$. Parabolic Schauder estimates (e.g. from the reference above or Lieberman's book) give $C^{3,\,\alpha}$ regularity of $v$, hence $C^{1,\alpha}$ regularity of $u$. $\endgroup$
    – Connor Mooney
    Commented Feb 21 at 21:19

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