Traveling wave solutions for an age-structured diffusion model in a shifting heterogeneous degenerate environment

We consider the age–structured diffusion equation ∂ t U + ∂ a U = d U x x + π ( a ) U 2 ( g ( x − c t ) − h ( x − c t ) U ) , t > 0 , x ∈ R , with the renewal condition U ( t , x , 0 ) = ∫ 0 A γ ( a ) U ( t , x , a ) d a , to describe a population with age–dependent reproduction and local saturation. Unlike the nondegenerate case studied in the pioneering work of Sherratt and Marchant [1], and unlike the age–structured model with Fisher–KPP nonlinearity considered by Ducrot [2], the reaction term here is degenerate, vanishing quadratically at extinction and cubically at saturation, and is coupled with an age structure. It is therefore challenging to analyze front propagation and to establish the monotonicity, uniqueness, and stability of traveling-wave solutions in this degenerate age–structured diffusion model, since the loss of linear behavior near extinction prevents linearization at the leading edge and invalidates the standard spectral characterization of a critical wave speed. Such degeneracy suppresses linear determinacy: the system admits traveling-wave solutions for all diffusion rates d > 0, without any critical propagation speed. Using an analytic semigroup framework together with age–dependent sub- and super-solutions, and relying on direct comparison and compactness arguments, we prove the existence, uniqueness, and spatial monotonicity of positive traveling-wave solutions connecting g − / h − and 0. The fronts are determined entirely by the nonlinear diffusion–reaction balance.