# variable coefficient heat equation

I am looking at $u_t=a(x,t)u_{xx}$ in D and with value=0 on the dD. $a(x,t)>a_0>0$. I want to find the requirements I need to impose on a(x,t) s.t. I have a strong stability, i.e. $||u(t,x)||^2\leq C*||u(x,0)||^2$ in some norm. I am doing that via energy methods using integration by parts a couple of times and get the following: $d||u^2||/dt=-(au_x,u_x)+(a_{xx}u,u)\leq -a_0(u_x,u_x)+(a_{xx}u,u)$. So this is where I am stuck. I can impose the condition so a(x,t) s.t. $a_{xx}\leq 0$ but this is very strong. Please suggest me something or a reference would be highly appreciated. I did look in a lot of books but can't find it. Thanks!

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Try looking at $d\|u_x\|^2/dt$ instead. –  Bob Pego Apr 17 '12 at 0:39
yes, I can see that that quantity is less than zero, but this is the estimate for the derivative in L^2, not for ||u|| and I did not impose any condition except positivity so far...how to proceed? –  Medan Apr 17 '12 at 1:55
edit: Poincare inequality implies that however still no additional restriction for a(x,t) rather than those for existence and uniqueness. But I guess that should be the case as uniqueness is somewhat connected to stability....? –  Medan Apr 17 '12 at 14:12