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| The parabolic partial differential equation $u_t = \Delta u - V(t,x)u + f (t,x,u) $ on $\mathbb{R}^N$ with a time $T$-periodic potential $V$ and nonlinearity $f$ shall be considered. The tail estimates method shall be applied to study the compactness properties of the translation along trajectories operator. By averaging arguments and fixed point techniques we prove that a topologically nontrivial stationary solution of $-\Delta u = \widehat V(x)u+ \widehat f (x,u)$, where $\widehat V$, $\widehat f$ are the time averages of $V$ and $f$, respectively, is the source of a branch of periodic solutions. Finally, by a continuation argument, we derive criteria for the existence of $T$-periodic solutions for asymptotically linear $f$ that interacts properly with the spectrum of $-\Delta + \widehat V$. |
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