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 and Continuous Dynamical Systems, 2017, 37 (2) : 915-944. doi: 10.3934/dcds.2017038

References:

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[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	A. Capella-Kort, Stable Solutions of Nonlinear Elliptic Equations: Qualitative and Regularity Properties, PhD thesis, Universitat Politècnica de Catalunya, 2005.
[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.
[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.
[15]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
[16]	E. Fašangová, Asymptotic analysis for a nonlinear parabolic equation on $\mathbb R$, Comment. Math. Univ. Carolinae, 39 (1998), 525--544.
[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.
[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.
[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.
[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.
[21]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc. , Englewood Cliffs, NJ, 1964.
[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.
[23]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
[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.
[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.
[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.
[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.
[28]	B. S. Kerner and V. V. Osipov, Autosolitons, Kluwer, Dordrecht, 1994.
[29]	E. H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 1997.
[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.
[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.
[32]	H.P. McKean, Nagumo's equation, Adv. Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X.
[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.
[34]	A. S. Mikhailov, Foundations of Synergetics, Springer-Verlag, Berlin, 1990.
[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.
[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.
[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.
[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.
[39]	J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
[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.
[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.
[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.
[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.
[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.
[45]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. , Birkhäuser Verlag, Basel, Switzerland, 2007.
[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.
[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.
[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.
[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.
[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.
[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.
[52]	J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	A. Capella-Kort, Stable Solutions of Nonlinear Elliptic Equations: Qualitative and Regularity Properties, PhD thesis, Universitat Politècnica de Catalunya, 2005.
[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.
[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.
[15]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
[16]	E. Fašangová, Asymptotic analysis for a nonlinear parabolic equation on $\mathbb R$, Comment. Math. Univ. Carolinae, 39 (1998), 525--544.
[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.
[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.
[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.
[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.
[21]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc. , Englewood Cliffs, NJ, 1964.
[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.
[23]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
[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.
[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.
[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.
[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.
[28]	B. S. Kerner and V. V. Osipov, Autosolitons, Kluwer, Dordrecht, 1994.
[29]	E. H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 1997.
[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.
[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.
[32]	H.P. McKean, Nagumo's equation, Adv. Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X.
[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.
[34]	A. S. Mikhailov, Foundations of Synergetics, Springer-Verlag, Berlin, 1990.
[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.
[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.
[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.
[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.
[39]	J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
[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.
[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.
[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.
[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.
[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.
[45]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher. , Birkhäuser Verlag, Basel, Switzerland, 2007.
[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.
[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.
[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.
[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.
[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.
[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.
[52]	J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230. doi: 10.1137/S0036144599364296.
[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.

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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2020 Impact Factor: 1.392

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2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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