Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity

By Yu-Juan Sun, Wan-Tong Li, Zhi-Cheng Wang. 2011

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Abstract This paper is concerned with traveling wave solutions of the equation ∂ u ∂ t = J ∗ u − u + f ( u ) on R × ( 0 , ∞ ) , where the dispersion kernel J is nonnegative and the nonlinearity f is monostable type. We show that there exists c ∗ ∈ R such that for any c > c ∗ , there is a nonincreasing traveling wave solution ϕ with ϕ ( − ∞ ) = 1 and lim ξ → ∞ ϕ ( ξ ) e λ ξ = 1 , where λ = Λ 1 ( c ) is the smallest positive solution to c λ = ∫ R J ( z ) e λ z d z − 1 + f ′ ( 0 ) . Furthermore, the existence of traveling wave solutions with c = c ∗ is also established. For c ≠ 0 , we further prove that the traveling wave solution is unique up to translation and is globally asymptotically stable in certain sense.

Subfields: Applied Mathematics, Mathematical Physics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/ba36579e19054b954ad128f5bb7d06dde29c90f1