Geodesic
Stabilization of Solution to the Cauchy Problem for a~Parabolic Equation with Lower Order Coefficients and an Exponentially Growing Initial Function
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 261 (2008), pp. 97-100

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For the coefficients of lower order terms of a second-order parabolic equation, we obtain sharp sufficient conditions under which the solution of the Cauchy problem stabilizes to zero uniformly in $x$ on each compact set $K$ in $\mathbb R^N$ for any exponentially growing initial function. 
@article{TM_2008_261_a7,
     author = {V. N. Denisov},
     title = {Stabilization of {Solution} to the {Cauchy} {Problem} for {a~Parabolic} {Equation} with {Lower} {Order} {Coefficients} and an {Exponentially} {Growing} {Initial} {Function}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {97--100},
     publisher = {mathdoc},
     volume = {261},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2008_261_a7/}
}
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V. N. Denisov. Stabilization of Solution to the Cauchy Problem for a~Parabolic Equation with Lower Order Coefficients and an Exponentially Growing Initial Function. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 261 (2008), pp. 97-100. http://geodesic.mathdoc.fr/item/TM_2008_261_a7/