American Institute of Mathematical Sciences

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

Grow up and slow decay in the critical Sobolev case

Marek Fila 1, and  John R. King 2,

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.
Keywords: asymptotic behaviour, Sobolev exponent., Semilinear heat equation.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B40, 35B3.
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.  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.   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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.   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.   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.  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.  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.  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.  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.  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.   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.  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.   Google Scholar

[27]

P. Quittner, The decay of global solutions of a semilinear heat equation,, Discrete Contin. Dynam. Systems A, 21 (2008), 307.  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, (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.   Google Scholar

[30]

C. Stinner, Very slow convergence to zero for a supercritical semilinear parabolic equation,, Adv. Differ. Equations, 14 (2009), 1085.   Google Scholar

[31]

C. Stinner, Very slow convergence rates in a semilinear parabolic equation,, NoDEA, 17 (2010), 213.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.   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.   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.  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.  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.  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.  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.  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.   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.  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.   Google Scholar

[27]

P. Quittner, The decay of global solutions of a semilinear heat equation,, Discrete Contin. Dynam. Systems A, 21 (2008), 307.  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, (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.   Google Scholar

[30]

C. Stinner, Very slow convergence to zero for a supercritical semilinear parabolic equation,, Adv. Differ. Equations, 14 (2009), 1085.   Google Scholar

[31]

C. Stinner, Very slow convergence rates in a semilinear parabolic equation,, NoDEA, 17 (2010), 213.  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.  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.  doi: 10.2307/2154232.  Google Scholar

[1]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365
[2]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[3]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159
[4]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388
[5]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[6]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[7]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[8]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
[9]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052
[10]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[11]

Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106
[12]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002
[13]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
[14]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
[15]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250
[16]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[17]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
[18]

Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286
[19]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282
[20]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences