Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

Cyrill B. Muratov; Xing Zhong

doi:10.3934/dcds.2017038

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February 2017, 37(2): 915-944. doi: 10.3934/dcds.2017038

Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

Cyrill B. Muratov and Xing Zhong ^1,

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA

¹The author is deceased.

Received June 2015 Revised January 2016 Published November 2016

Fund Project: This work was supported, in part, by NSF via grants DMS-0908279, DMS-1119724 and DMS-1313687. CBM wishes to express his gratitude to V. Moroz for many valuable discussions.

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in $\mathbb{R}^N$ with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For $L^2$ initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.

Keywords: Sharp transition, traveling waves, gradient flow.

Mathematics Subject Classification: Primary:35K57, 35K15, 35A1.

Citation: Cyrill B. Muratov, Xing Zhong. Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 915-944. doi: 10.3934/dcds.2017038

References:

[1]	S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal., 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2. Google Scholar
[2]	D.G. Aronson and H.F. Weinberger, Multidimensional diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar
[3]	R. Bamón, I. Flores and M. del Pino, Ground states of semilinear elliptic equations: A geometric approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 551-581. doi: 10.1016/S0294-1449(00)00126-8. Google Scholar
[4]	P.W. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264. doi: 10.1016/S0022-1236(02)00013-7. Google Scholar
[5]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. Google Scholar
[6]	H. Berestycki, P.-L. Lions and L.A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb R^n$, Indiana Univ. Math. J., 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012. Google Scholar
[7]	G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on ${\bf R}^n$ or ${\bf R}^n_{+}$ through the method of moving planes, Comm. Partial Differential Equations, 22 (1997), 1671-1690. doi: 10.1080/03605309708821315. Google Scholar
[8]	J. Busca, M.A. Jendoubi and P. Poláčik, Convergence to equilibrium for semilinear parabolic problems in $\mathbb{R}^n$, Comm. Partial Differential Equations, 27 (2002), 1793-1814. doi: 10.1081/PDE-120016128. Google Scholar
[9]	X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $\mathbb{R}^n$, C. R. Math. Acad. Sci. Paris, 338 (2004), 769-774. doi: 10.1016/j.crma.2004.03.013. Google Scholar
[10]	X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093. Google Scholar
[11]	L.A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar
[12]	A. Capella-Kort, Stable Solutions of Nonlinear Elliptic Equations: Qualitative and Regularity Properties, PhD thesis, Universitat Politècnica de Catalunya, 2005. Google Scholar
[13]	E.N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., 131 (2003), 1891-1899. doi: 10.1090/S0002-9939-02-06733-3. Google Scholar
[14]	Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312. doi: 10.4171/JEMS/198. Google Scholar
[15]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. Google Scholar
[16]	E. Fašangová, Asymptotic analysis for a nonlinear parabolic equation on $\mathbb R$, Comment. Math. Univ. Carolinae, 39 (1998), 525--544. Google Scholar
[17]	E. Feireisl, On the long time behaviour of solutions to nonlinear diffusion equations on Rⁿ, Nonlin. Diff. Eq. Appl., 4 (1997), 43-60. doi: 10.1007/PL00001410. Google Scholar
[18]	E. Feireisl and H. Petzeltová, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential Integral Equations, 10 (1997), 181-196. Google Scholar
[19]	P.C. Fife, Long time behavior of solutions of bistable nonlinear diffusion equations, Arch. Rational Mech. Anal., 70 (1979), 31-46. doi: 10.1007/BF00276380. Google Scholar
[20]	J. Földes and P. Poláčik, Convergence to a steady state for asymptotically autonomous semilinear heat equations on $\mathbb{R}^n$, J. Differential Equations, 251 (2011), 1903-1922. doi: 10.1016/j.jde.2011.04.002. Google Scholar
[21]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc. , Englewood Cliffs, NJ, 1964. Google Scholar
[22]	V.A. Galaktionov, S.I. Pokhozhaev and A.E. Shishkov, On convergence in gradient systems with a degenerate equilibrium position, Mat. Sb., 198 (2007), 65-88. doi: 10.1070/SM2007v198n06ABEH003862. Google Scholar
[23]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. Google Scholar
[24]	C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in Rⁿ, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906. Google Scholar
[25]	C.K. R.T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364. doi: 10.1216/RMJ-1983-13-2-355. Google Scholar
[26]	C.K. R.T. Jones, Spherically symmetric solutions of a reaction-diffusion equation, J. Diff. Equations, 49 (1983), 142-169. doi: 10.1016/0022-0396(83)90023-2. Google Scholar
[27]	Y.I. Kanel', On the stabilization of solutions of the Cauchy problem for the equations arising in the theory of combusion, Mat. Sbornik, 59 (1962), 245-288. Google Scholar
[28]	B. S. Kerner and V. V. Osipov, Autosolitons, Kluwer, Dordrecht, 1994. Google Scholar
[29]	E. H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 1997. Google Scholar
[30]	C.S. Lin and W.-M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Math. Soc., 102 (1988), 271-277. doi: 10.1090/S0002-9939-1988-0920985-9. Google Scholar
[31]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Basel, 1995. Google Scholar
[32]	H.P. McKean, Nagumo's equation, Adv. Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar
[33]	A.G. Merzhanov and E.N. Rumanov, Physics of reaction waves, Rev. Mod. Phys., 71 (1999), 1173-1210. doi: 10.1103/RevModPhys.71.1173. Google Scholar
[34]	A. S. Mikhailov, Foundations of Synergetics, Springer-Verlag, Berlin, 1990. Google Scholar
[35]	C.B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 867-892. doi: 10.3934/dcdsb.2004.4.867. Google Scholar
[36]	C.B. Muratov and M. Novaga, Front propagation in infinite cylinders. I. A variational approach, Comm. Math. Sci., 6 (2008), 799-826. doi: 10.4310/CMS.2008.v6.n4.a1. Google Scholar
[37]	C.B. Muratov and M. Novaga, Global stability and exponential convergence to variational traveling waves in cylinders, SIAM J. Math. Anal., 44 (2012), 293-315. doi: 10.1137/110833269. Google Scholar
[38]	C.B. Muratov and X. Zhong, Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations, Nonlin. Diff. Eq. Appl., 20 (2013), 1519-1552. doi: 10.1007/s00030-013-0220-7. Google Scholar
[39]	J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. Google Scholar
[40]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IEEE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. Google Scholar
[41]	P. Poláčik, Morse indices and bifurcations of positive solutions of $Δ u+f(u)=0$ on $\mathbb{R}^n$, Indiana Univ. Math. J., 50 (2001), 1407-1432. doi: 10.1512/iumj.2001.50.1909. Google Scholar
[42]	P. Poláčik and K.P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations, 124 (1996), 472-494. doi: 10.1006/jdeq.1996.0020. Google Scholar
[43]	P. Poláčik and E. Yanagida, Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics, SIAM J. Math. Anal., 46 (2014), 3481-3496. doi: 10.1137/140958566. Google Scholar
[44]	P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on $\mathbb{R}^n$, Arch. Ration. Mech. Anal., 199 (2011), 69-97. doi: 10.1007/s00205-010-0316-8. Google Scholar
[45]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. , Birkhäuser Verlag, Basel, Switzerland, 2007. Google Scholar
[46]	V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 341-379. doi: 10.1016/S0294-1449(03)00042-8. Google Scholar
[47]	J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893. Google Scholar
[48]	J. Shi and X. Wang, Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations, J. Differential Equations, 231 (2006), 235-251. doi: 10.1016/j.jde.2006.06.008. Google Scholar
[49]	L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals Math., 118 (1983), 525-571. doi: 10.2307/2006981. Google Scholar
[50]	M. Tang, Existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations, J. Differential Equations, 164 (2000), 155-179. doi: 10.1006/jdeq.1999.3752. Google Scholar
[51]	K. Uchiyama, Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients, Arch. Rational Mech. Anal., 90 (1985), 291-311. doi: 10.1007/BF00276293. Google Scholar
[52]	J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296. Google Scholar
[53]	A. Zlatoš, Sharp transition between extinction and propagation of reaction, J. Amer. Math. Soc., 19 (2006), 251-263. doi: 10.1090/S0894-0347-05-00504-7. Google Scholar

show all references

References:

[1]	S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal., 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2. Google Scholar
[2]	D.G. Aronson and H.F. Weinberger, Multidimensional diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar
[3]	R. Bamón, I. Flores and M. del Pino, Ground states of semilinear elliptic equations: A geometric approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 551-581. doi: 10.1016/S0294-1449(00)00126-8. Google Scholar
[4]	P.W. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264. doi: 10.1016/S0022-1236(02)00013-7. Google Scholar
[5]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. Google Scholar
[6]	H. Berestycki, P.-L. Lions and L.A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb R^n$, Indiana Univ. Math. J., 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012. Google Scholar
[7]	G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on ${\bf R}^n$ or ${\bf R}^n_{+}$ through the method of moving planes, Comm. Partial Differential Equations, 22 (1997), 1671-1690. doi: 10.1080/03605309708821315. Google Scholar
[8]	J. Busca, M.A. Jendoubi and P. Poláčik, Convergence to equilibrium for semilinear parabolic problems in $\mathbb{R}^n$, Comm. Partial Differential Equations, 27 (2002), 1793-1814. doi: 10.1081/PDE-120016128. Google Scholar
[9]	X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $\mathbb{R}^n$, C. R. Math. Acad. Sci. Paris, 338 (2004), 769-774. doi: 10.1016/j.crma.2004.03.013. Google Scholar
[10]	X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093. Google Scholar
[11]	L.A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar
[12]	A. Capella-Kort, Stable Solutions of Nonlinear Elliptic Equations: Qualitative and Regularity Properties, PhD thesis, Universitat Politècnica de Catalunya, 2005. Google Scholar
[13]	E.N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., 131 (2003), 1891-1899. doi: 10.1090/S0002-9939-02-06733-3. Google Scholar
[14]	Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312. doi: 10.4171/JEMS/198. Google Scholar
[15]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. Google Scholar
[16]	E. Fašangová, Asymptotic analysis for a nonlinear parabolic equation on $\mathbb R$, Comment. Math. Univ. Carolinae, 39 (1998), 525--544. Google Scholar
[17]	E. Feireisl, On the long time behaviour of solutions to nonlinear diffusion equations on Rⁿ, Nonlin. Diff. Eq. Appl., 4 (1997), 43-60. doi: 10.1007/PL00001410. Google Scholar
[18]	E. Feireisl and H. Petzeltová, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential Integral Equations, 10 (1997), 181-196. Google Scholar
[19]	P.C. Fife, Long time behavior of solutions of bistable nonlinear diffusion equations, Arch. Rational Mech. Anal., 70 (1979), 31-46. doi: 10.1007/BF00276380. Google Scholar
[20]	J. Földes and P. Poláčik, Convergence to a steady state for asymptotically autonomous semilinear heat equations on $\mathbb{R}^n$, J. Differential Equations, 251 (2011), 1903-1922. doi: 10.1016/j.jde.2011.04.002. Google Scholar
[21]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc. , Englewood Cliffs, NJ, 1964. Google Scholar
[22]	V.A. Galaktionov, S.I. Pokhozhaev and A.E. Shishkov, On convergence in gradient systems with a degenerate equilibrium position, Mat. Sb., 198 (2007), 65-88. doi: 10.1070/SM2007v198n06ABEH003862. Google Scholar
[23]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. Google Scholar
[24]	C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in Rⁿ, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906. Google Scholar
[25]	C.K. R.T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math., 13 (1983), 355-364. doi: 10.1216/RMJ-1983-13-2-355. Google Scholar
[26]	C.K. R.T. Jones, Spherically symmetric solutions of a reaction-diffusion equation, J. Diff. Equations, 49 (1983), 142-169. doi: 10.1016/0022-0396(83)90023-2. Google Scholar
[27]	Y.I. Kanel', On the stabilization of solutions of the Cauchy problem for the equations arising in the theory of combusion, Mat. Sbornik, 59 (1962), 245-288. Google Scholar
[28]	B. S. Kerner and V. V. Osipov, Autosolitons, Kluwer, Dordrecht, 1994. Google Scholar
[29]	E. H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 1997. Google Scholar
[30]	C.S. Lin and W.-M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Math. Soc., 102 (1988), 271-277. doi: 10.1090/S0002-9939-1988-0920985-9. Google Scholar
[31]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Basel, 1995. Google Scholar
[32]	H.P. McKean, Nagumo's equation, Adv. Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar
[33]	A.G. Merzhanov and E.N. Rumanov, Physics of reaction waves, Rev. Mod. Phys., 71 (1999), 1173-1210. doi: 10.1103/RevModPhys.71.1173. Google Scholar
[34]	A. S. Mikhailov, Foundations of Synergetics, Springer-Verlag, Berlin, 1990. Google Scholar
[35]	C.B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 867-892. doi: 10.3934/dcdsb.2004.4.867. Google Scholar
[36]	C.B. Muratov and M. Novaga, Front propagation in infinite cylinders. I. A variational approach, Comm. Math. Sci., 6 (2008), 799-826. doi: 10.4310/CMS.2008.v6.n4.a1. Google Scholar
[37]	C.B. Muratov and M. Novaga, Global stability and exponential convergence to variational traveling waves in cylinders, SIAM J. Math. Anal., 44 (2012), 293-315. doi: 10.1137/110833269. Google Scholar
[38]	C.B. Muratov and X. Zhong, Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations, Nonlin. Diff. Eq. Appl., 20 (2013), 1519-1552. doi: 10.1007/s00030-013-0220-7. Google Scholar
[39]	J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. Google Scholar
[40]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IEEE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. Google Scholar
[41]	P. Poláčik, Morse indices and bifurcations of positive solutions of $Δ u+f(u)=0$ on $\mathbb{R}^n$, Indiana Univ. Math. J., 50 (2001), 1407-1432. doi: 10.1512/iumj.2001.50.1909. Google Scholar
[42]	P. Poláčik and K.P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations, 124 (1996), 472-494. doi: 10.1006/jdeq.1996.0020. Google Scholar
[43]	P. Poláčik and E. Yanagida, Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics, SIAM J. Math. Anal., 46 (2014), 3481-3496. doi: 10.1137/140958566. Google Scholar
[44]	P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on $\mathbb{R}^n$, Arch. Ration. Mech. Anal., 199 (2011), 69-97. doi: 10.1007/s00205-010-0316-8. Google Scholar
[45]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. , Birkhäuser Verlag, Basel, Switzerland, 2007. Google Scholar
[46]	V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 341-379. doi: 10.1016/S0294-1449(03)00042-8. Google Scholar
[47]	J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893. Google Scholar
[48]	J. Shi and X. Wang, Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations, J. Differential Equations, 231 (2006), 235-251. doi: 10.1016/j.jde.2006.06.008. Google Scholar
[49]	L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals Math., 118 (1983), 525-571. doi: 10.2307/2006981. Google Scholar
[50]	M. Tang, Existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations, J. Differential Equations, 164 (2000), 155-179. doi: 10.1006/jdeq.1999.3752. Google Scholar
[51]	K. Uchiyama, Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients, Arch. Rational Mech. Anal., 90 (1985), 291-311. doi: 10.1007/BF00276293. Google Scholar
[52]	J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296. Google Scholar
[53]	A. Zlatoš, Sharp transition between extinction and propagation of reaction, J. Amer. Math. Soc., 19 (2006), 251-263. doi: 10.1090/S0894-0347-05-00504-7. Google Scholar

Table 1. List of critical exponents.

Name	Exponent	Validity	$N = 3$
Fujita	$p_F = (N + 2)/N$	$N \geq 1$	5/3
Serrin	$p_{sg} = N / (N - 2)$	$N \geq 3$	3
Sobolev	$p_S = (N + 2) / (N - 2)$	$N \geq 3$	5
Joseph-Lundgren	$p_{JL} = 1 + 4/ \left( N - 4 - 2 \sqrt{N - 1} \, \right)$	$N \geq 11$	-

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Name	Exponent	Validity	$N = 3$
Fujita	$p_F = (N + 2)/N$	$N \geq 1$	5/3
Serrin	$p_{sg} = N / (N - 2)$	$N \geq 3$	3
Sobolev	$p_S = (N + 2) / (N - 2)$	$N \geq 3$	5
Joseph-Lundgren	$p_{JL} = 1 + 4/ \left( N - 4 - 2 \sqrt{N - 1} \, \right)$	$N \geq 11$	-

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