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We consider the second order parabolic partial differential equation    ${\sum }_{i,j=1}^n{a}_{ij}(x,t)u_{x_i{x}_j}+{\sum }_{i=1}^n{b}_i(x,t)u_{x_i}+c(x,t)u-u_t=0$. Sufficient conditions are given under which every solution of the above equation must decay or tend to infinity as |x|→ ∞. A sufficient condition is also given under which every solution of a system of the form    $L^{\alpha }\left[u^{\alpha }\right]+{\sum }_{\beta =1}^N{c}^{\alpha \beta }(x,t)u^{\beta }=f^{\alpha }(x,t)$, where    $L^{\alpha }\left[u\right]\equiv {\sum }_{i,j=1}^n{a}_{ij}^{\alpha }(x,t)u_{x_i{x}_j}+{\sum }_{i=1}^n{b}_i^{\alpha }(x,t)u_{x_i}-u_t$, must decay as t → ∞.