Dear Math Overflowers,

I'm looking for references on the parabolic Harnack inequality for distributional solutions of the heat equation on the whole space $$ \partial_t u=\Delta u\quad\text{and}\quad u(0)=u_0\quad\Leftrightarrow\quad u(t,x)=\int\limits_{R^d}\Gamma_t(x-y)u_0(y)dy. $$ Here $\Gamma_t(z)=\frac{e^{-\frac{|z|^2}{4t}}}{(4\pi t)^{d/2}}$ is the usual heat kernel and my initial data $u_0\geq 0$ is only in $L^1\cap L^p(R^d)$ for some $p>1$.

More precisely, I'm interested in the case when the initial datum $u_0$ is supported on a positive finite measure set $E=supp(u_0)$ and $u_0|_E\geq M>0$ for some constant $M$. I would like to control $u(t,x)\geq (\ldots)$ for points $x\in supp(u_0)$ and small times $t>0$, which really looks like Harnack inequality to me. The thing is that I have no information at all on the initial support (except that it has finite measure so it may not even be bounded) and I only have mere $L^1\cap L^p$ regularity for $u_0$. I strongly doubt that I can get pointwise estimates, but maybe $L^p$ estimates for $|u(t)|_{L^p(E)}\geq (\ldots)$?

Thank you in advance!


Without knowing how the initial data is "spread," we can't get either pointwise or $L^p$ estimates independent of the measure of the initial support. Take for example the building block solution $v$ with initial data $\chi_{B_{\epsilon}}$. It follows from the representation formula that at time $t = \epsilon$, $v \leq C\epsilon^{n/2}$ on $B_{\epsilon}$. If we add together $\epsilon^{-n}$ translations of $v$ sufficiently "spread apart" (depending on $\epsilon$) to get $u$, it is easy to see (by exponential spatial decay) that the same bound will hold at all the points where $u$ was initially supported, which has measure like $1$.

No pointwise estimate follows immediately. To see no $L^p$, if we compute at $t = \epsilon$ we get $$\int_{E} u^p \leq C\epsilon^{-n} \int_{B_{\epsilon}} \epsilon^{np/2} \leq C\epsilon^{np/2}.$$

  • $\begingroup$ great answer @Connor Mooney, thanks a lot! But darn, I whish you were wrong ;-) $\endgroup$ – leo monsaingeon Oct 24 '13 at 16:22

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