Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line

ABSTRACT

We consider the accelerated propagation of solutions to equations with a nonlocal linear dispersion on the real line and monostable nonlinearities (both local or nonlocal, however, not degenerated at 0), in the case when either of the dispersion kernel or the initial condition has regularly heavy tails at both $\pm \infty$, perhaps different. We show that, in such case, the propagation to the right direction is fully determined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone initial conditions which may give different orders of the acceleration. Our approach is based, in particular, on the extension of the theory of sub-exponential distributions, which we introduced early.

KEYWORDS:

• Nonlocal diffusion
• reaction–diffusion equation
• front propagation
• acceleration
• monostable equation
• nonlocal nonlinearity
• long-time behaviour
• integral equation

AMS SUBJECT CLASSIFICATIONS:

• 35B40
• 35K57
• 47G20
• 45G10

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Authors gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft through CRC 701 “Stochastic Dynamics: Mathematical Theory and Applications” (DF and PT), by the European Commission through the project FP7-PEOPLE-2013-IRSES-612669 “STREVCOMS” (DF), and by the “Bielefeld Young Researchers” Fund through the Funding Line Postdocs: “Career Bridge Doctorate – Postdoc” (PT).