The proof is by Moser iteration: a technique that extracts pointwise bounds from $L^p$ estimates by testing the equation with powers of the solution and iterating the resulting Sobolev inequalities. The strategy has three stages: (1) derive a local $L^p$-to-$L^\infty$ bound for subsolutions (the "sup estimate"), (2) derive a local $L^{-p}$-to-$L^{-\infty}$ bound for non-negative supersolutions (the "inf estimate"), and (3) combine them using an intermediate $L^p$ estimate that links the two time levels. **Step 1: The sup estimate (Moser's forward iteration).** [claim:Local Sup Bound For Subsolutions] If $u_t + Lu \le 0$ in $U_T$ and $B(x_0, 2R) \subset U$, then for any $0 < s < t \le T$: \begin{align*} \sup_{B(x_0, R) \times [t - (t-s)/2, \, t]} u^+ \le C \left(\frac{1}{R^n (t - s)} \int_s^t \int_{B(x_0, 2R)} (u^+)^2 \, d\mathcal{L}^n \, d\tau\right)^{1/2}. \end{align*} [/claim] [proof] Test the inequality $u_t + Lu \le 0$ with $v = \zeta^2 u |u|^{p-2}$ for $p \ge 2$, where $\zeta$ is a smooth cutoff [function](/page/Function) supported in a parabolic cylinder $Q_R := B(x_0, R) \times (s, t)$. The time [derivative](/page/Derivative) term gives $\frac{1}{p}\frac{d}{dt}\int \zeta^2 |u|^p$, and the elliptic part gives a coercive contribution of order $\int \zeta^2 |\nabla(|u|^{p/2})|^2$. After absorbing cross terms via Young's inequality, one obtains \begin{align*} \sup_{s < \tau < t} \int \zeta^2 |u|^p \, d\mathcal{L}^n + \int_s^t \int \zeta^2 |\nabla(|u|^{p/2})|^2 \, d\mathcal{L}^n \, d\tau \le C(p) \int_s^t \int (|\nabla \zeta|^2 + |\zeta_t|) |u|^p. \end{align*} The Sobolev embedding $H^1(B) \hookrightarrow L^{2^*}(B)$ (with $2^* = 2n/(n-2)$ for $n \ge 3$) then gives \begin{align*} \|u\|_{L^{p \cdot (1 + 2/n)}(Q_{R/2})} \le C(p, R)^{1/p} \|u\|_{L^p(Q_R)}. \end{align*} Setting $\chi = 1 + 2/n > 1$ and iterating: $p_0 = 2$, $p_1 = 2\chi$, $p_2 = 2\chi^2$, $\ldots$, $p_k = 2\chi^k \to \infty$. The product of constants $\prod_{k=0}^\infty C(p_k, R)^{1/p_k}$ converges (a geometric series in the exponent), giving the $L^2$-to-$L^\infty$ bound. [/proof] **Step 2: The inf estimate (Moser's backward iteration).** [claim:Local Inf Bound For Non-Negative Supersolutions] If $u \ge 0$ and $u_t + Lu \ge 0$ in $U_T$, and $B(x_0, 2R) \subset U$, then for any $0 < s < t \le T$: \begin{align*} \inf_{B(x_0, R) \times [s, \, s + (t-s)/2]} u \ge C^{-1} \left(\frac{1}{R^n(t-s)} \int_s^t \int_{B(x_0, 2R)} u^{-q} \, d\mathcal{L}^n \, d\tau\right)^{-1/q} \end{align*} for some $q > 0$. [/claim] [proof] Test with $v = \zeta^2 u^{-(p+1)}$ for $p > 0$ (which is valid since $u > 0$ by the [Strong Maximum Principle (Parabolic)](/theorems/613) or a limiting argument). The resulting inequality, after the same Sobolev iteration as in Step 1 but with negative exponents ($p_k = -2\chi^k \to -\infty$), yields a bound on $\|u^{-1}\|_{L^\infty}$ in terms of $\|u^{-1}\|_{L^q}$, which is equivalent to a lower bound on $\inf u$ in terms of a negative-power $L^q$ average. [/proof] **Step 3: The linking estimate.** The critical step is to show that the $L^2$ average at the later time $t_2$ controls the $L^{-q}$ average at the earlier time $t_1$. This uses the **Bombieri–Giusti logarithmic estimate**: for a non-negative solution $u$ of $u_t + Lu = 0$, the function $w := \log u$ satisfies a differential inequality that allows one to bound $\int e^{\pm \alpha w}$ for small $\alpha > 0$, linking positive and negative $L^p$ norms of $u$. The key identity is \begin{align*} (\log u)_t + L(\log u) = -\frac{|\nabla u|^2}{u^2} \sum_{i,j} a_{ij} \le 0, \end{align*} so $\log u$ is a subsolution. A Poincaré-type inequality for $\log u$ on parabolic cylinders, combined with the John–Nirenberg lemma, shows that $\log u$ has bounded mean oscillation, which gives \begin{align*} \left(\frac{1}{|Q|}\int_Q u^q\right)^{1/q} \le C \left(\frac{1}{|Q|}\int_Q u^{-q}\right)^{-1/q} \end{align*} for some $q > 0$ and parabolic cylinders $Q$ with the correct time ordering ($Q$ at the earlier time for $u^{-q}$, at the later time for $u^q$). **Step 4: Assembly.** For a non-negative solution $u$ of $u_t + Lu = 0$, combining Steps 1–3: \begin{align*} \sup_{V \times \{t_1\}} u &\le C_1 \|u\|_{L^2(Q_1)} & &\text{(sup estimate at earlier time)} \\ &\le C_2 \|u^{-1}\|_{L^q(Q_2)}^{-1} & &\text{(linking estimate)} \\ &\le C_3 \inf_{V \times \{t_2\}} u & &\text{(inf estimate at later time)}, \end{align*} where $Q_1$ and $Q_2$ are appropriately chosen parabolic cylinders containing $V \times \{t_1\}$ and $V \times \{t_2\}$ respectively, with $t_1 < t_2$. The constant $C = C_1 C_2 C_3$ depends on $V$, $t_1$, $t_2$, $\theta$, $\|a_{ij}\|_{L^\infty}$, and $n$.