Let $ u_0\in L^\infty (\mathbb{R^d})\cap C^\alpha(\mathbb{R}^d), \alpha \in (0,1), \Phi $ the fundamental solution of the heat equation and $u(t,\cdot ):= \Phi (t,\cdot )*u_0(\cdot)$ for $t>0$. Let $ T,R >0$. Then for every $t\in (0,T), x\in\mathbb{R}^d$ with $|x|<R$, and $i,j\in \{ 1,...,d\} $

$$ |\partial_{ij} u(t,x)|\leq \frac{c}{t^{1-\frac{\alpha}{2}}},$$

with $ c\geq 1 $ constant.

Can someone give me hints?

  • $\begingroup$ a good place to start would be to write down $u$ $\endgroup$ – qbert Jan 5 '18 at 12:02
  • $\begingroup$ $$u(t,x)= \frac{1}{(4\pi t)^{(d/2)}}\int_{\mathbb{R}^d} exp(-\frac{|x-y|^2}{4t})u_0(y) dy$$ and i know that $ \partial_{ij}u=\partial_{ij}\Phi*u_0$, i can calculate this. But then i dont know how to use that $u_0\in C^\alpha$. $\endgroup$ – user494831 Jan 5 '18 at 13:28
  • $\begingroup$ Try doing the computation in the Fourier domain. The condition $u_0 \in C^\alpha$ tells you something about the decay of the Fourier transform of $u_0$. $\endgroup$ – Jeff Jan 5 '18 at 22:20
  • $\begingroup$ Sorry, but what is the Fourier domain? We dont speak about the Fourier transormation in the lectures, only about convolution. $\endgroup$ – user494831 Jan 6 '18 at 9:14

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