It is embarrassing that I have no idea if this is true or not, even in 1D. Consider a parabolic PDE in $\mathbb{R}$ $$ \partial_t u - a(x,t)\partial_{xx} u + b(x,t) \partial_x u + c(x)u = 0 \,,$$ With, say, $a > 0,b,c$ bounded smooth functions and non-negative smooth $L^2(\mathbb{R})$ initial data. If $c(x) > \gamma >0$, then by comparison principle, the solution to this equation converges exponentially fast to $0$ as $t \to \infty$. Suppose now we relax the condition for $c(x) > 0$ to, say, $c(x) \to 0$ algebraically as $|x| \to \infty$, is it true that there exists a function $g(x) >0$ such that $$ u(x,t) \leq C e^{-g(x)t} ?$$
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