New results on traveling wave solutions for a Keller-Segel system with nonlinear chemical gradient

We study a Keller-Segel system with nonlinear chemical gradient and two parameters c > 0 and ε > 0. The system has a family of equilibria (u^*,v_±^*) with u₁^**2^* where u₁^* and u₂^* are explicitly defined. We investigate the existence of traveling wave solutions (u_s^c,ε(x-st),u_s^c,ε(x-st)) of this system that connect a pair of equilibria (u^*,v_±^*) and establish the following result: there exists u₀^* with u₁^*0^*2^* such that if u₁^**≤u₀^*, then (u_s^c,ε,v_s^c,ε) exists for all c > 0 and s > 0 and sufficiently small ε > 0; if u₀^**2^*, then there exists c* > 0 such that for c > c*, such a solution exists for all s > 0 and sufficiently small ε, while for 0 < c ≤ c*, such a solution exists only for s in a disconnected set of (0, ∞) which includes two connected components (0,s_c^*) and (s^_c,∞), where 0c^*c<∞, s_c^* is increasing and s^_c decreasing in c, [Formula: see text] and [Formula: see text] . This result generalizes the main result in [J. Math. Anal. Appl., 545 (2025), 129128], where a special case c=ε in the system was studied.