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Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants.

Let $Q(R) := Q(R^2,R) = B(0, R) \times [-R^2,0] \subset \mathbb R^d$ be the standard parabolic cylinder. Suppose $u$ is a nonnegative solution of the heat equation $$\tag1 u_t - div[a(x,t)\nabla u] =0 \quad \text{in}\ \ Q(2R), $$ with a Hölder continuous coefficient $a(x,t)$ satisfying $0<C_o \le a(x,t) \le C_1$ for some constants $C_o, C_1>0$. Now the well-known Hölder estimate tells us that if $$ \sup_{Q(R)} u \le M_o \in \mathbb R_+, $$ then $$ osc_{Q(r)} u \le \gamma_oM_o\left(\frac rR\right)^{\alpha_o}, $$ for all $0 <r \le R$ and for some constants $\gamma_o, \alpha_o$ which depend only on $C_o, C_1$ and the dimension $d$.

Now my question concerns the scaling properties of this result. Namely, let $v$ be a nonnegative solution of (1) in $Q(2A^\frac12R),\ A\ge1,$ and assume $$\tag2 \sup_{Q(AR^2, R)} v \le M_1, $$ for some constant $M_1 \in \mathbb R_+$, i.e. we assume the upper bound in a cylinder stretched in time direction (while $v$ is a solution in the larger cylinder $Q(2A^\frac12R)$ which has the standard parabolic scaling). My question is: does this imply $$ osc_{Q(Ar^2,r)} v \le \gamma_1M_1\left(\frac rR\right)^{\alpha_1}, \quad 0 < r \le R, $$ for constants $\gamma_1$ and $\alpha_1$ independent of $A$? It is quite easy to prove this for constants depending on $A$, but the question is whether such a dependence is necessary. I suspect that this does not hold, in general, but since I am not very good in finding counter-examples, I would ask your help for this.

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I don't think that oscillation decay estimates in the long cylinders are independent of $A$. Indeed, if we let $A^{1/2}R = 1$ so we are working in $Q_1$, and then take $A$ to infinity, this is like saying we have oscillation decay in time at the spatial origin.

However, if we take boundary data $\sin(Bt)$ and solve the heat equation, I think that asymptotically the values at the origin are like $\cos(Bt)/B$, so if we took $M_1$ instead to be the the oscillation of $v$ in time at the origin (of order $1/B$) we would violate the inequality for $B$ large. (This is a counterexample to oscillation decay in the long thin cylinders, but not the inequality as stated where we have $M_1$ or order $1$).

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