Abstract
Motivated by the uniqueness problem for monostable semi-wave-fronts, we propose a revised version of the Diekmann and Kaper theory of a nonlinear convolution equation. Our version of the Diekmann–Kaper theory allows (1) to consider new types of models which include nonlocal KPP type equations (with either symmetric or anisotropic dispersal), nonlocal lattice equations and delayed reaction–diffusion equations; (2) to incorporate the critical case (which corresponds to the slowest wavefronts) into the consideration; (3) to weaken or to remove various restrictions on kernels and nonlinearities. The results are compared with those of Schumacher (J Reine Angew Math 316: 54–70, 1980), Carr and Chmaj (Proc Am Math Soc 132: 2433–2439, 2004), and other more recent studies.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aguerrea M., Trofimchuk S., Valenzuela G.: Uniqueness of fast travelling fronts in reaction–diffusion equations with delay. Proc. R. Soc. A 464, 2591–2608 (2008)
Bates P., Fife P., Ren X., Wang X.: Traveling waves in a convolution model for phase transitions. Arch. Ration. Mech. Anal. 138, 105–136 (1997)
Berestycki H., Nirenberg L.: Traveling waves in cylinders. Ann. Inst. H. Poincare Anal. Non. Lineaire 9, 497–572 (1992)
Carr J., Chmaj A.: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc 132, 2433–2439 (2004)
Chen X., Fu S.-C., Guo J.-S.: Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices. SIAM J. Math. Anal 38, 233–258 (2006)
Chen X., Guo J.-S.: Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math. Ann 326, 123–146 (2003)
Coville J.: On uniqueness and monotonicity of solutions of nonlocal reaction diffusion equation. Ann. Mat. Pura Appl 185, 461–485 (2006)
Coville J., Dávila J., Martínez S.: Nonlocal anisotropic dispersal with monostable nonlinearity. J. Differ. Equ. 244, 3080–3118 (2008)
Coville J., Dupaigne L.: On a non-local equation arising in population dynamics. Proc. Roy. Soc. Edinburgh Sect. A 137, 727–755 (2007)
Diekmann, O.: On a nonlinear integral equation arising in mathematical epidemiology. In: Eckhaus, W., de Jager, E.M. (eds.) Differential Equations and Applications (Proc. Third Scheveningen Conf.), Scheveningen, 1977, pp. 133–140. North-Holland, Amsterdam (1978)
Diekmann O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol 6, 109–130 (1978)
Diekmann O., Kaper H.: On the bounded solutions of a nonlinear convolution equation. Nonlinear Anal. TMA 2, 721–737 (1978)
Ebert U., van Saarloos W.: Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Phys. D 146, 1–99 (2000)
Engel K.-J., Nagel R.: One-parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Ermentrout G., McLeod J.: Existence and uniqueness of traveling waves for a neural network. Proc. Roy. Soc. Edinburg Sect. A 123, 461–478 (1993)
Fang J., Wei J., Zhao X.-Q.: Uniqueness of traveling waves for nonlocal lattice equations. Proc. Amer. Math. Soc 139, 1361–1373 (2011)
Fang J., Zhao X.: Existence and uniqueness of traveling waves for non-monotone integral equations with applications. J. Differ. Equ. 248, 2199–2226 (2010)
Faria T., Huang W., Wu J.: Traveling waves for delayed reaction–diffusion equations with non-local response. Proc. R. Soc. A 462, 229–261 (2006)
Faria T., Trofimchuk S.: Non-monotone traveling waves in a single species reaction–diffusion equation with delay. J. Differ. Equ. 228, 357–376 (2006)
Faria T., Trofimchuk S.: Positive travelling fronts for reaction–diffusion systems with distributed delay. Nonlinearity 23, 2457–2481 (2010)
Garnier J.: Accelerating solutions in integro-differential equations. SIAM J. Math. Anal. 43, 1955–1974 (2011)
Gilding B., Kersner R.: Travelling Waves in Nonlinear Diffusion–Convection Reaction. Birkhauser, Basel (2004)
Gomez A., Trofimchuk S.: Monotone traveling wavefronts of the KPP-Fisher delayed equation. J. Differ. Equ. 250, 1767–1787 (2011)
Gourley S., So J., Wu J.: Non-locality of reaction–diffusion equations induced by delay: biological modeling and nonlinear dynamics. J. Math. Sci. 124, 5119–5153 (2004)
Guo J.-S., Hamel F.: Front propagation for discrete periodic monostable equations. Math. Ann. 335, 489–525 (2006)
Guo J.-S., Wu C.-H.: Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system. Osaka J. Math 45, 327–346 (2008)
Hale J., Verduyn Lunel S.: Introduction to Functional Differential Equations. Springer, New York (1993)
Kolmogorov A., Petrovskii I., Piskunov N.: Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem. Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh 1, 1–26 (1937)
Ma S.: Traveling waves for non-local delayed diffusion equations via auxiliary equations. J. Differ. Equ. 237, 259–277 (2007)
Ma S., Zou X.: Existence, uniqueness and stability of travelling waves in a discrete reaction–diffusion monostable equation with delay. J. Differ. Equ. 217, 54–87 (2005)
Mallet-Paret J.: The Fredholm alternative for functional differential equations of mixed type. J. Dynam. Differ. Equ. 11, 1–48 (1999)
Mei M., Lin C.-K., Lin C-T., So J.W.-H.: Traveling wavefronts for time-delayed reaction–diffusion equation, (II) nonlocal nonlinearity. J. Differ. Equ. 247, 511–529 (2009)
Schumacher K.: Travelling-front solutions for integro-differential equations. I. J. Reine Angew. Math. 316, 54–70 (1980)
Stokes A.: On two types of moving front in quasilinear diffusion. Math. Biosci. 31, 307–315 (1976)
Titchmarsh E.: Introduction to the Theory of Fourier Integrals. Chelsea, New York (1986)
Thieme H., Zhao X.-Q.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction–diffusion models. J. Differ. Equ. 195, 430–470 (2003)
Trofimchuk E., Tkachenko V., Trofimchuk S.: Slowly oscillating wave solutions of a single species reaction–diffusion equation with delay. J. Differ. Equ. 245, 2307–2332 (2008)
Trofimchuk E., Alvarado P., Trofimchuk S.: On the geometry of wave solutions of a delayed reaction–diffusion equation. J. Differ. Equ. 246, 1422–1444 (2009)
Wang Z.-C., Li W.T., Ruan S.: Traveling fronts in monostable equations with nonlocal delayed effects. J. Dynam. Differ. Equ. 20, 573–607 (2008)
Widder D.: The Laplace Transform. Princeton University Press, Princeton (1941)
Yagisita H.: Existence and Nonexistence of Travelling Waves for a Nonlocal Monostable Equation, vol. 45, pp. 925–953. Publication of RIMS Kyoto University, Kyoto (2009)
Zinner B., Harris G., Hudson W.: Traveling wavefronts for the discrete Fisher’s equation. J. Differ. Equ. 105, 46–62 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aguerrea, M., Gomez, C. & Trofimchuk, S. On uniqueness of semi-wavefronts. Math. Ann. 354, 73–109 (2012). https://doi.org/10.1007/s00208-011-0722-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-011-0722-8