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Abstract

We consider the second order parabolic partial differential equation    $∑^n_{i,j=1} a_{ij}(x,t) u_{x_{i}x_{j}} + ∑^n_{i=1} b_i(x,t) u_{x_i} + c(x,t)u - u_t = 0$. 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    $L^α[u^α] + ∑_{β=1}^N c^{αβ}(x,t) u^β = f^α(x,t)$, where    $L^α[u] ≡ ∑^n_{i,j=1} a_{ij}^α(x,t) u_{x_{i}x_{j}} + ∑^n_{i=1} b_i^α(x,t) u_{x_i} - u_t$, must decay as t → ∞.