= Heat-equation uniqueness under Gaussian growth
{title2=$|u(x,t)|\le M e^{a|x|^2}$}
Classical <heat equation> solutions with identical <continuous> initial data and a uniform Gaussian spatial growth bound on each finite time interval are unique. For the zero-data difference, choose $b>a$ and compare on a short slab with $\varepsilon\Phi_b$, where $\Phi_b=(1-4bt)^{-n/2}\exp(b|x|^2/(1-4bt))$ solves the equation for $t<1/(4b)$. Its faster spatial growth controls the lateral boundary of large cylinders. The <heat equation maximum principle> bounds both signs of the difference by $\varepsilon\Phi_b$ inside. Increasing the cylinder radius and then sending $\varepsilon$ to zero proves uniqueness on that slab; repeating slabs covers the finite interval.
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