Traveling Waves for a Sign-Changing Nonlocal Evolution Equation with Delayed Nonlocal Response

This paper deals with the existence of traveling wave solutions for a nonlocal evolution equation with delayed nonlocal response and sign-changing kernel. By constructing a new pair of upper-lower solutions and applying Schauder’s fixed point theorem, we first prove that there exists a number \(c^{\#}>0\) $c^{\#}>0$ such that when \(c>c^{\#}\) $c>c^{\#}$ , the nonlocal evolution equation admits a semi-wave solution with wave speed c, which connects the trivial equilibrium 0 at negative infinity. Then, we analyze the asymptotic behavior of wave profile at positive infinity and obtain the existence of a traveling wave solution with speed c and connecting the trivial equilibrium 0 and the positive equilibrium 1, when the wave speed c is large.