THE PERIODIC-PARABOLIC LOGISTIC EQUATION ON RN

In this article, we investigate the periodic-parabolic logistic equation on the entire space R-N (N >= 1): {partial derivative(t)u - Delta u = a(x, t)u - b(x, t)u(p) in R-N x (0, T), u(x, 0) = u(x, T) in R-N, where the constants T > 0 and p > 1, and the functions a, b with b > 0 are smooth in R-N x R and T-periodic in time. Under the assumptions that a(x, t)/vertical bar x vertical bar(gamma) and b(x, t)/vertical bar x vertical bar(tau) are bounded away from 0 and infinity for all large vertical bar x vertical bar, where the constants gamma > -2 and tau is an element of R, we study the existence and uniqueness of positive T-periodic solutions. In particular, we determine the asymptotic behavior of the unique positive T-periodic solution as vertical bar x vertical bar -> infinity, which turns out to depend on the sign of gamma. Our investigation considerably generalizes the existing results.