# American Institute of Mathematical Sciences

December  2012, 7(4): 661-671. doi: 10.3934/nhm.2012.7.661

## Grow up and slow decay in the critical Sobolev case

 1 Department of Applied Mathematics and Statistics, Comenius University, 84248 Bratislava, Slovak Republic 2 Division of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, United Kingdom

Received  January 2012 Revised  May 2012 Published  December 2012

We present conjectures on asymptotic behaviour of threshold solutions of the Cauchy problem for a semilinear heat equation with Sobolev critical nonlinearity. The conjectures say that, depending on the decay rate of initial data and the space dimension, the threshold solutions may grow up, stabilize, or decay to zero as $t→∞$. The rates of grow up or decay are computed formally using matched asymptotics.
Citation: Marek Fila, John R. King. Grow up and slow decay in the critical Sobolev case. Networks & Heterogeneous Media, 2012, 7 (4) : 661-671. doi: 10.3934/nhm.2012.7.661
##### References:
 [1] M. Fila, J. R. King, M. Winkler and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. Differ. Equations, 228 (2006), 339-356. doi: 10.1016/j.jde.2006.01.019.  Google Scholar [2] M. Fila, J. R. King, M. Winkler and E. Yanagida, Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent, Adv. Differ. Equations, 12 (2007), 1-26.  Google Scholar [3] M. Fila, J. R. King, M. Winkler and E. Yanagida, Very slow grow-up of solutions of a semi-linear parabolic equation, Proc. Edinb. Math. Soc., 53 (2011), 1-20. doi: 10.1017/S0013091509001497.  Google Scholar [4] M. Fila, H. Matano and P. Poláčik, Immediate regularization after blow-up, SIAM J. Math. Anal., 37 (2005), 752-776. doi: 10.1137/040613299.  Google Scholar [5] M. Fila and M. Winkler, Rate of convergence to a singular steady state of a supercritical parabolic equation, J. Evol. Equations, 8 (2008), 673-692. doi: 10.1007/s00028-008-0400-9.  Google Scholar [6] M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. Differ. Equations, 205 (2004), 365-389. doi: 10.1016/j.jde.2004.03.009.  Google Scholar [7] M. Fila, M. Winkler and E. Yanagida, Convergence rate for a parabolic equation with supercritical nonlinearity, J. Dynam. Differ. Equations, 17 (2005), 249-269. doi: 10.1007/s10884-005-5405-2.  Google Scholar [8] M. Fila, M. Winkler and E. Yanagida, Slow convergence to zero for a parabolic equation with supercritical nonlinearity, Math. Annalen, 340 (2008), 477-496. doi: 10.1007/s00208-007-0148-5.  Google Scholar [9] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.  Google Scholar [10] V. A. Galaktionov and J. R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differ. Equations, 189 (2003), 199-233. doi: 10.1016/S0022-0396(02)00151-1.  Google Scholar [11] V. Galaktionov and J. L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Applied Math., 50 (1997), 1-67. doi: 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R.  Google Scholar [12] C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbbR^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906.  Google Scholar [13] C. Gui, W.-M. Ni and X. Wang, Further study on a nonlinear heat equation, J. Differ. Equations, 169 (2001), 588-613. doi: 10.1006/jdeq.2000.3909.  Google Scholar [14] M. Hoshino and E. Yanagida, Sharp estimates of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity, Nonlin. Anal. TMA, 69 (2008), 3136-3152. doi: 10.1016/j.na.2007.09.007.  Google Scholar [15] R. Ikehata, M. Ishiwata and T. Suzuki, Semilinear parabolic equation in $R^N$ associated with critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 69 (2010), 877-900. doi: 10.1016/j.anihpc.2010.01.002.  Google Scholar [16] M. Ishiwata, On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent, J. Differ. Equations, 249 (2010), 1466-1482. doi: 10.1016/j.jde.2010.06.024.  Google Scholar [17] O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423-452.  Google Scholar [18] T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 1-15.  Google Scholar [19] H. Matano and F. Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 256 (2009), 992-1064. doi: 10.1016/j.jfa.2008.05.021.  Google Scholar [20] H. Matano and F. Merle, Threshold and generic type I behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 261 (2011), 716-748. doi: 10.1016/j.jfa.2011.02.025.  Google Scholar [21] N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity, Math. Z., 239 (2002), 215-219. doi: 10.1007/s002090100292.  Google Scholar [22] N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its applications, Indiana Univ. Math. J., 54 (2005), 1047-1059. doi: 10.1512/iumj.2005.54.2694.  Google Scholar [23] W.-M. Ni, P. E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differ. Equations, 54 (1984), 97-120. doi: 10.1016/0022-0396(84)90145-1.  Google Scholar [24] P. Poláčik and P. Quittner, Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation, Asymptotic Analysis, 57 (2008), 125-141.  Google Scholar [25] P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Annalen, 327 (2003), 745-771. doi: 10.1007/s00208-003-0469-y.  Google Scholar [26] P. Poláčik and E. Yanagida, Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation, Diff. Int. Equations, 17 (2004), 535-548.  Google Scholar [27] P. Quittner, The decay of global solutions of a semilinear heat equation, Discrete Contin. Dynam. Systems A, 21 (2008), 307-318. doi: 10.3934/dcds.2008.21.307.  Google Scholar [28] P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar [29] Ph. Souplet, Sur l'asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domaines non bornés, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 877-882.  Google Scholar [30] C. Stinner, Very slow convergence to zero for a supercritical semilinear parabolic equation, Adv. Differ. Equations, 14 (2009), 1085-1106.  Google Scholar [31] C. Stinner, Very slow convergence rates in a semilinear parabolic equation, NoDEA, 17 (2010), 213-227. doi: 10.1007/s00030-009-0050-9.  Google Scholar [32] C. Stinner, The convergence rate for a semilinear parabolic equation with a critical exponent, Appl. Math. Letters, 24 (2011), 454-459. doi: 10.1016/j.aml.2010.10.041.  Google Scholar [33] X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-589. doi: 10.2307/2154232.  Google Scholar

show all references

##### References:
 [1] M. Fila, J. R. King, M. Winkler and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. Differ. Equations, 228 (2006), 339-356. doi: 10.1016/j.jde.2006.01.019.  Google Scholar [2] M. Fila, J. R. King, M. Winkler and E. Yanagida, Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent, Adv. Differ. Equations, 12 (2007), 1-26.  Google Scholar [3] M. Fila, J. R. King, M. Winkler and E. Yanagida, Very slow grow-up of solutions of a semi-linear parabolic equation, Proc. Edinb. Math. Soc., 53 (2011), 1-20. doi: 10.1017/S0013091509001497.  Google Scholar [4] M. Fila, H. Matano and P. Poláčik, Immediate regularization after blow-up, SIAM J. Math. Anal., 37 (2005), 752-776. doi: 10.1137/040613299.  Google Scholar [5] M. Fila and M. Winkler, Rate of convergence to a singular steady state of a supercritical parabolic equation, J. Evol. Equations, 8 (2008), 673-692. doi: 10.1007/s00028-008-0400-9.  Google Scholar [6] M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. Differ. Equations, 205 (2004), 365-389. doi: 10.1016/j.jde.2004.03.009.  Google Scholar [7] M. Fila, M. Winkler and E. Yanagida, Convergence rate for a parabolic equation with supercritical nonlinearity, J. Dynam. Differ. Equations, 17 (2005), 249-269. doi: 10.1007/s10884-005-5405-2.  Google Scholar [8] M. Fila, M. Winkler and E. Yanagida, Slow convergence to zero for a parabolic equation with supercritical nonlinearity, Math. Annalen, 340 (2008), 477-496. doi: 10.1007/s00208-007-0148-5.  Google Scholar [9] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.  Google Scholar [10] V. A. Galaktionov and J. R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differ. Equations, 189 (2003), 199-233. doi: 10.1016/S0022-0396(02)00151-1.  Google Scholar [11] V. Galaktionov and J. L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Applied Math., 50 (1997), 1-67. doi: 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R.  Google Scholar [12] C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbbR^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906.  Google Scholar [13] C. Gui, W.-M. Ni and X. Wang, Further study on a nonlinear heat equation, J. Differ. Equations, 169 (2001), 588-613. doi: 10.1006/jdeq.2000.3909.  Google Scholar [14] M. Hoshino and E. Yanagida, Sharp estimates of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity, Nonlin. Anal. TMA, 69 (2008), 3136-3152. doi: 10.1016/j.na.2007.09.007.  Google Scholar [15] R. Ikehata, M. Ishiwata and T. Suzuki, Semilinear parabolic equation in $R^N$ associated with critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 69 (2010), 877-900. doi: 10.1016/j.anihpc.2010.01.002.  Google Scholar [16] M. Ishiwata, On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent, J. Differ. Equations, 249 (2010), 1466-1482. doi: 10.1016/j.jde.2010.06.024.  Google Scholar [17] O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423-452.  Google Scholar [18] T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 1-15.  Google Scholar [19] H. Matano and F. Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 256 (2009), 992-1064. doi: 10.1016/j.jfa.2008.05.021.  Google Scholar [20] H. Matano and F. Merle, Threshold and generic type I behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 261 (2011), 716-748. doi: 10.1016/j.jfa.2011.02.025.  Google Scholar [21] N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity, Math. Z., 239 (2002), 215-219. doi: 10.1007/s002090100292.  Google Scholar [22] N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its applications, Indiana Univ. Math. J., 54 (2005), 1047-1059. doi: 10.1512/iumj.2005.54.2694.  Google Scholar [23] W.-M. Ni, P. E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differ. Equations, 54 (1984), 97-120. doi: 10.1016/0022-0396(84)90145-1.  Google Scholar [24] P. Poláčik and P. Quittner, Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation, Asymptotic Analysis, 57 (2008), 125-141.  Google Scholar [25] P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Annalen, 327 (2003), 745-771. doi: 10.1007/s00208-003-0469-y.  Google Scholar [26] P. Poláčik and E. Yanagida, Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation, Diff. Int. Equations, 17 (2004), 535-548.  Google Scholar [27] P. Quittner, The decay of global solutions of a semilinear heat equation, Discrete Contin. Dynam. Systems A, 21 (2008), 307-318. doi: 10.3934/dcds.2008.21.307.  Google Scholar [28] P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar [29] Ph. Souplet, Sur l'asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domaines non bornés, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 877-882.  Google Scholar [30] C. Stinner, Very slow convergence to zero for a supercritical semilinear parabolic equation, Adv. Differ. Equations, 14 (2009), 1085-1106.  Google Scholar [31] C. Stinner, Very slow convergence rates in a semilinear parabolic equation, NoDEA, 17 (2010), 213-227. doi: 10.1007/s00030-009-0050-9.  Google Scholar [32] C. Stinner, The convergence rate for a semilinear parabolic equation with a critical exponent, Appl. Math. Letters, 24 (2011), 454-459. doi: 10.1016/j.aml.2010.10.041.  Google Scholar [33] X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-589. doi: 10.2307/2154232.  Google Scholar
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