heat equation, regularity

From: Peter Spellucci (spellucci_at_fb04373.mathematik.tu-darmstadt.de)
Date: 07/01/04 

Date: Thu, 1 Jul 2004 14:04:20 +0000 (UTC)

dear community,

in a numerical analysis book I found without reference and without proof the
following used as "obvious"

 let
 (d/d t) u = (d/d_x) (a(x) (d/d x) u ) for x in R and t>0
 and
 u(x,0) = u0(x), x in R with u0 in C_0^4(R) .
 further let
 a in C^{infty}(R) , a(x)>0 for x in R and a' in C_0^{infty}(R)
 (hence a is bounded from below by a positive constant and is bounded globally,
  indeed constant for |x| large enough. for the question C^4 would be clearly
  enough).

 then
 for any t>0 (d/d x)^4 (a(x)u(x,t)) is in L_2(R).
 (this is obvious for constant a)
 but how to prove that for nonconstant a??
 any hint or literature is highly appreciated
 thanks
 p. spellucci

 

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