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I am trying estimates the space derivatives of solution of heat equation in infinity norm with bounded initial data in $\mathbb{R^3}$

Here is the equation

$u_t=\Delta u$ in $ B(0,1)\times (0,T) $

$u=0$ in $(0,T]\times \partial B(0,1)$

$u(x,0)=f$ in $B(0,1)$.

Where $f\in l^{\infty}(B(0,1))$ and $B(0,1)$ is unit sphere.

I tried to find the solution using the $C^{\infty}$ cut off function but could not quite get the solution in term of the initial data. I could not find any reference about it as well.I would be happy to get some reference that talks about the heat kernel or solution of heat equation for a bounded domain in $\mathbb{R^n}$.

Bret lee
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    You can epress $\Delta u$ in spherical coordinates. Assuming spherical symmetry the stady state equation is simple. – Emilio Novati Jun 25 '17 at 16:39
  • If the $\times$ indicates a Cartesian product then, for my information, what does $\mbox{} \times \partial B(0,1)$ mean? – thb Jun 25 '17 at 16:40
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    @thb vertical surface with out top and bottom.Think of a cylinder. – Bret lee Jun 25 '17 at 16:42
  • I realize that you have come to ask a question rather than to answer one, but may I ask: out of what kind of book would one learn such notation? Multivariable analysis? Something else? I ask because I have seen such notation but never an explanation of it. – thb Jun 25 '17 at 16:52
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    @thb: in this context $\partial$ is the ''boundary operator'': https://en.wikipedia.org/wiki/Boundary_%28topology%29 – Emilio Novati Jun 25 '17 at 17:00
  • @Bretlee: If the initial distribution $f$ has spherical symmetry you can find a solution to the transient problem here: http://www.ewp.rpi.edu/hartford/~ernesto/S2006/CHT/Notes/ch03.pdf – Emilio Novati Jun 25 '17 at 17:03
  • @EmilioNovati, I think the problem is the boundary data f which is just bounded . Anyway thanks for your comments. – Bret lee Jun 25 '17 at 17:28

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