Monotone traveling waves for the viscous Nicholson's blowflies equation

This paper studies the existence and stability of monotone traveling wave solutions for the viscous Nicholson's blowflies equation$ \begin{align*} u_t(t, x)-ku_{xxt}(t, x) = D_mu_{xx}(t, x)-d_mu(t, x)+\epsilon pu(t-\tau, x) e^{-au(t-\tau, x)}, \end{align*} $where $ 0 < k \leq \frac{D_m}{d_m} $, $ 1 < \frac{\epsilon p}{d_m} \leq e $, and $ \tau \geq 0 $. There exists a minimal wave speed $ c^*(k, \tau) $, and we prove the existence of decreasing traveling wave solutions for $ c>c^*(k, \tau) $ using the monotone iteration technique combined with the upper-lower solutions method. A limiting argument is then employed to establish the critical case $ c = c^*(k, \tau) $. Furthermore, we demonstrate the non-existence of traveling wave solutions when $ c<c^*(k, \tau) $. In particular, without time delay (i.e., $ \tau = 0 $), $ c^*(k, 0) $ increases as $ k $ increases. For all $ c > c^*(k, \tau) $, the global exponential stability of the traveling waves is established using the weighted energy method.