On uniqueness of traveling waves for a reaction diffusion equation with spatio-temporal delay

Spreading speeds in an asymptotic autonomous system with application to an epidemic model

“…is an upper solution to (3.1) if A 5 > 1 such that the initial condition holds (see Appendix A). When the (IL1) and (IR1) is true, we can refer to the asymptotic behavior of traveling waves in delayed equations (see a recent work by Xu & Xiao [40]) to obtain an upper solution. For other cases, the conclusion can be verified by different combinations of these cases.…”

“…$$ When (IL2) and (IR2) hold, we find $$$$ \left({A}_5\min \left\{{e}^{\gamma_{c_l}\left(x+{c}_lt\right)},{e}^{\gamma_{c_r}\left(-x+{c}_rt\right)}\Big\},{A}_5\min \Big\{{\eta}_{c_l}{e}^{\gamma_{c_l}\left(x+{c}_lt\right)},{\eta}_{c_r}{e}^{\gamma_{c_r}\left(-x+{c}_rt\right)}\right\}\right) $$$$ is an upper solution to () if $$$ {A}_5>1 $$$ such that the initial condition holds (see Appendix ). When the (IL1) and (IR1) is true, we can refer to the asymptotic behavior of traveling waves in delayed equations (see a recent work by Xu & Xiao [40]) to obtain an upper solution. For other cases, the conclusion can be verified by different combinations of these cases.…”