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evinda

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I want to prove that the solution $u$, $u(x,t)=\frac{1}{2 \sqrt{k \pi t}} \int_{-\infty}^{+\infty} e^{-\frac{(x-s)^2}{4kt}} \phi(s) ds, x \in \mathbb{R}, t>0 (\star)$, of the initial value problem for the heat equation, with continuous and bounded initial value $\phi$, is infinitely many times differentiable, for $t>0$.

We suppose that $|\phi(s)| \leq M$.

I found that $\sup_{x \in \mathbb{R}} |u_t(x,t)| \leq C_1 M \frac{1}{t}$ for some constant $C_1$ and $\sup_{x \in \mathbb{R}}|u_x(x,t)| \leq c_2 M \frac{1}{\sqrt{t}}$, for some constant $c_2$.

At the hint it says that we continue showing that all the derivatives of $u$ exist, for a positive $t$, using the following proposition:

Let $\lambda$ be a positive number and $n$ a natural number. Then $\int_{-\infty}^{+\infty} x^{2n} e^{-2 \lambda x^2} dx<+\infty$. How do we use this proposition? (Thinking)