Fujita blow up phenomena and hair trigger effect: The role of dispersal tails

Alfaro, Matthieu

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1309-1327.

We consider the nonlocal diffusion equation ${\partial }_{t}u=J⁎u-u+u^{1+p}$ in the whole of ${\mathbb{R}}^N$. We prove that the Fujita exponent dramatically depends on the behavior of the Fourier transform of the kernel J near the origin, which is linked to the tails of J. In particular, for compactly supported or exponentially bounded kernels, the Fujita exponent is the same as that of the nonlinear Heat equation ${\partial }_{t}u=\mathrm{\Delta }u+u^{1+p}$. On the other hand, for kernels with algebraic tails, the Fujita exponent is either of the Heat type or of some related Fractional type, depending on the finiteness of the second moment of J. As an application of the result in population dynamics models, we discuss the hair trigger effect for ${\partial }_{t}u=J⁎u-u+u^{1+p}(1-u)$.

[1] Alfaro, M. Slowing Allee effect vs. accelerating heavy tails in monostable reaction diffusion equations (submitted for publication) | arXiv | MR | Zbl

[2] M. Alfaro, J. Coville, Propagation phenomena in monostable integro-differential equations: acceleration or not?, submitted for publication. | MR

[3] Aronson, D.G.; Weinberger, H.F. Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., Volume 30 (1978), pp. 33–76 | DOI | MR | Zbl

[4] Bebernes, J.W.; Li, C.; Li, Y. Travelling fronts in cylinders and their stability, Rocky Mt. J. Math., Volume 27 (1997), pp. 123–150 | DOI | MR | Zbl

[5] Birkner, M.; López-Mimbela, J.A.; Wakolbinger, A. Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach, Proc. Am. Math. Soc., Volume 130 (2002), pp. 2431–2442 (electronic) | DOI | MR | Zbl

[6] Chasseigne, E.; Chaves, M.; Rossi, J.D. Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9), Volume 86 (2006), pp. 271–291 | DOI | MR | Zbl

[7] Deng, K.; Levine, H.A. The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., Volume 243 (2000), pp. 85–126 | DOI | MR | Zbl

[8] Durrett, R. Probability: Theory and Examples, Duxbury Press, Belmont, CA, 1996 | MR

[9] Fisher, R.A. The wave of advance of advantageous genes, Annu. Eugen., Volume 7 (1937), pp. 355–369 | JFM

[10] Fujita, H. On the blowing up of solutions of the Cauchy problem for $u_t=\mathrm{\Delta }u+u^{1+\alpha }$ , J. Fac. Sci., Univ. Tokyo, Sect. I, Volume 13 (1966), pp. 109–124 | MR | Zbl

[11] García-Melián, J.; Quirós, F. Fujita exponents for evolution problems with nonlocal diffusion, J. Evol. Equ., Volume 10 (2010), pp. 147–161 | DOI | MR | Zbl

[12] García-Melián, J.; Rossi, J.D. On the principal eigenvalue of some nonlocal diffusion problems, J. Differ. Equ., Volume 246 (2009), pp. 21–38 | DOI | MR | Zbl

[13] Guedda, M.; Kirane, M. A note on nonexistence of global solutions to a nonlinear integral equation, Bull. Belg. Math. Soc. Simon Stevin, Volume 6 (1999), pp. 491–497 | DOI | MR | Zbl

[14] Gui, C.; Huan, T. Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc. Var. Partial Differ. Equ., Volume 54 (2015), pp. 251–273 | MR | Zbl

[15] Hayakawa, K. On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Jpn. Acad., Volume 49 (1973), pp. 503–505 | MR | Zbl

[16] Kaplan, S. On the growth of solutions of quasi-linear parabolic equations, Commun. Pure Appl. Math., Volume 16 (1963), pp. 305–330 | DOI | MR | Zbl

[17] Kobayashi, K.; Sirao, T.; Tanaka, H. On the growing up problem for semilinear heat equations, J. Math. Soc. Jpn., Volume 29 (1977), pp. 407–424 | DOI | MR | Zbl

[18] Kolmogorov, A.N.; Petrovsky, I.G.; Piskunov, N.S. Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Etat Moscou, Sér. Inter. A, Volume 1 (1937), pp. 1–26 | Zbl

[19] Levine, H.A. The role of critical exponents in blowup theorems, SIAM Rev., Volume 32 (1990), pp. 262–288 | DOI | MR | Zbl

[20] Mellet, A.; Roquejoffre, J.-M.; Sire, Y. Existence and asymptotics of fronts in non local combustion models, Commun. Math. Sci., Volume 12 (2014), pp. 1–11 | DOI | MR | Zbl

[21] Mitidieri, È.; Pokhozhaev, S.I. A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, Volume 234 (2001), pp. 1–384 | MR | Zbl

[22] Quittner, P.; Souplet, P. Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts. Basl. Lehrb., Birkhäuser Verlag, Basel, 2007 | MR | Zbl

[23] Sugitani, S. On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., Volume 12 (1975), pp. 45–51 | MR | Zbl

[24] Weissler, F.B. Existence and nonexistence of global solutions for a semilinear heat equation, Isr. J. Math., Volume 38 (1981), pp. 29–40 | DOI | MR | Zbl

[25] Xin, J. Existence and nonexistence of traveling waves and reaction–diffusion front propagation in periodic media, J. Stat. Phys., Volume 73 (1993), pp. 893–926 | MR | Zbl

[26] Yang, J.; Zhou, S.; Zheng, S. Asymptotic behavior of the nonlocal diffusion equation with localized source, Proc. Am. Math. Soc., Volume 142 (2014), pp. 3521–3532 | DOI | MR | Zbl

[27] Zhang, G.-B.; Li, W.-T.; Wang, Z.-C. Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differ. Equ., Volume 252 (2012), pp. 5096–5124 | MR | Zbl

[28] Zlatoš, A. Quenching and propagation of combustion without ignition temperature cutoff, Nonlinearity, Volume 18 (2005), pp. 1463–1475 | DOI | MR | Zbl