  This paper concerns a diffusive logistic equation with a free boundary and
sign-changing intrinsic growth rate in heterogeneous time-periodic environment,
in which the variable intrinsic growth rate may be "very negative" in a
"suitable large region". Such a model can be used to describe the spreading of
a new or invasive species, with the free boundary representing the expanding
front. In the case of higher space dimensions with radial symmetry and the
intrinsic growth rate has a positive lower bound, this problem has been studied
by Du, Guo and Peng . They established a spreading-vanishing dichotomy, the
sharp criteria for spreading and vanishing and estimate of the asymptotic
spreading speed. In the present paper, we show that the above results are
retained for our problem. This paper has been submitted to Journal of
Functional Analysis in August 5, 2014 (JFA-14-548).
