{'fraction_non_alphanumeric': 0.052795031055900624, 'fraction_numerical': 0.012422360248447204, 'mean_word_length': 4.758928571428571, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Bifurcation from Zero of a Complete Trajectory for nonautonomous logistic PDEs

In this paper we extend the well-known bifurcation theory for autonomous logistic equations to the nonautonomous equation \[ u_t-\Delta u=\lambda u-b(t)u^2 \quad \mbox{with} \quad b(t)\in[b_0,B_0], \] 0 < b0 < B0 < 2b0. In particular, we prove the existence of a unique uniformly bounded trajectory that bifurcates from zero as λ passes through the first eigenvalue of the Laplacian, which attracts all other trajectories. Although it is this relatively simple equation that we analyze in detail, other more involved models can be treated using similar techniques.