Journal of Elliptic and Parabolic Equations

, Volume 5, Issue 2, pp 423–471 | Cite as

Accelerated front propagation for monostable equations with nonlocal diffusion: multidimensional case

  • Dmitri FinkelshteinEmail author
  • Yuri Kondratiev
  • Pasha Tkachov


We describe acceleration of the front propagation for solutions to a class of monostable nonlinear equations with a nonlocal diffusion in \({{\mathbb {R}}^d}\), \(d\ge 1\). We show that the acceleration takes place if either the diffusion kernel or the initial condition has ‘regular’ heavy tails in \({{\mathbb {R}}^d}\) (in particular, decays slower than exponentially). Under general assumptions which can be verified for particular models, we present sharp estimates for the time-space zone which separates the region of convergence to the unstable zero solution with the region of convergence to the stable positive constant solution. We show the variety of different possible rates of the propagation starting from a little bit faster than a linear one up to the exponential rate. The paper generalizes to the case \(d>1\) our results for the case \(d=1\) obtained early in Finkelshtein and Tkachov (Appl Anal 98(4):756–780, 2019).


Nonlocal diffusion Reaction–diffusion equation Front propagation Acceleration Monostable equation Nonlocal nonlinearity Long-time behavior Integral equation 

Mathematics Subject Classification

35B40 35K57 47G20 45G10 



No funding was received.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.


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© Orthogonal Publisher and Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsSwansea UniversitySwanseaUK
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  3. 3.Gran Sasso Science InstituteL’AquilaItaly

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