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Akira
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Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x (\sigma^2 (u (t, x)) u(t, x))$

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 $$ \partial_t u (t, x) = \Delta_x (\sigma^2 (u (t, x)) u(t, x)). $$

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

  • $u_0 \in C^\infty_b (\mathbb R^d)$ is a probability density function.
  • 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!

Akira
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