TY - JOUR

T1 - The Bramson delay in a Fisher–KPP equation with log-singular nonlinearity

AU - Bouin, Emeric

AU - Henderson, Christopher

N1 - Funding Information:
CH was partially supported by National Science Foundation, United States of America grant DMS-2003110 .
Publisher Copyright:
© 2021 Elsevier Ltd

PY - 2021/12

Y1 - 2021/12

N2 - We consider a class of reaction–diffusion equations of Fisher–KPP type in which the nonlinearity (reaction term) f is merely C1 at u=0 due to a logarithmic competition term. We first derive the asymptotic behavior of (minimal speed) traveling wave solutions that is, we obtain precise estimates on the decay to zero of the traveling wave profile at infinity. We then use this to characterize the Bramson shift between the traveling wave solutions and solutions of the Cauchy problem with localized initial data. We find a phase transition depending on how singular f is near u=0 with quite different behavior for more singular f. This is in contrast to the smooth case, that is, when f∈C1,δ, where these behaviors are completely determined by f′(0). In the singular case, several scales appear and require new techniques to understand.

AB - We consider a class of reaction–diffusion equations of Fisher–KPP type in which the nonlinearity (reaction term) f is merely C1 at u=0 due to a logarithmic competition term. We first derive the asymptotic behavior of (minimal speed) traveling wave solutions that is, we obtain precise estimates on the decay to zero of the traveling wave profile at infinity. We then use this to characterize the Bramson shift between the traveling wave solutions and solutions of the Cauchy problem with localized initial data. We find a phase transition depending on how singular f is near u=0 with quite different behavior for more singular f. This is in contrast to the smooth case, that is, when f∈C1,δ, where these behaviors are completely determined by f′(0). In the singular case, several scales appear and require new techniques to understand.

KW - Logarithmic delay

KW - Reaction–diffusion equations

KW - Traveling waves

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U2 - 10.1016/j.na.2021.112508

DO - 10.1016/j.na.2021.112508

M3 - Article

AN - SCOPUS:85112362940

SN - 0362-546X

VL - 213

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

M1 - 112508

ER -