Fronts and patterns with a dynamic parameter ramp

The paper’s Theorem 1 proves that for FKPP with a linearly increasing parameter µ(t)=εt (with µ0=0) and Heaviside initial data, one has lim_{t→∞} sup_{x≥σ(t)+ηt} u(x,t)=0 and lim_{t→∞} inf_{x≤σ(t)−ηt} u(x,t)=1, where σ(t)=(4t∫_0^t µ(s)ds)^{1/2}=√(2ε) t^{3/2} (, ). The paper establishes the upper bound using a supersolution of the linearized equation (Lemma 5.1) and the lower bound via a specially constructed difference-of-exponentials subsolution in a comoving frame (, ). The candidate solution proves the same two limits by a different, simpler route: (i) a direct comparison with the linearized solution and Gaussian tail bounds for the leading edge, and (ii) a complementary barrier for v=1−u using the linear equation w_t=w_xx−µ(t)w with exponential initial majorant, yielding uniform convergence to 1 behind the front. The logic matches the theorem’s claims and uses standard heat-kernel calculations consistent with the paper’s linear predictions of σ(t) and the instantaneous steepness ν(t)=−σ(t)/(2t) (, ). Minor notes: the paper’s displayed closed-form for the linear supersolution includes an (x+σ(t)) shift inside an error function; while the derivation from the heat kernel yields an erfc(x/(2√t)) factor, this discrepancy amounts to a notational/placement choice for emphasizing level sets and does not affect the stated limits. The candidate solution states a comparison lemma with a bounded coefficient; one sentence explaining the standard integrating-factor reduction (removing the unbounded µ(t) on finite time intervals) would close that small gap. Overall, both are correct, with different proof strategies.