Forced periodic solutions for nonresonant parabolic equations on R^N

Criteria for the existence of $T$-periodic solutions of nonautonomous parabolic equation $u_t = \Delta u + f(t,x,u)$, $x\in\mathbb{R}^N$, $t>0$ with asymptotically linear $f$ will be provided. It is expressed in terms of time average function $\hat f$ of the nonlinear term $f$ and the spectrum of the Laplace operator $\Delta$ on $\mathbb{R}^N$. One of them says that if the derivative $\hat f_\infty$ of $\hat f$ at infinity does not interact with the spectrum of $\Delta$, i.e. $\mathrm{Ker} (-\Delta + \hat f_\infty)=\{0\}$, then the parabolic equation admits a $T$-periodic solution. Another theorem is derived in the situation, where the linearization at $0$ and infinity differ topologically, i.e. the total multiplicities of negative eigenvalues of the averaged linearizations at $0$ and $\infty$ are different mod $2$.