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Filtering the $ L^2- $critical focusing Schrödinger equation
Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity
1. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China |
2. | Institute of Applied Physics and Computational Mathematics, Beijing 100088, China |
$ u_t = u_{xx}+x^m|u_x|^p, \ \ t>0, \ \ 0<x<1, $ |
$ p>m+2 $ |
$ m\geq0 $ |
$ u $ |
$ u $ |
$ C^1 $ |
$ u_x $ |
References:
[1] |
N. D. Alikakos, P. W. Bates and C. P. Grant,
Blow up for a diffusion-advection equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 181-190.
doi: 10.1017/S0308210500024057. |
[2] |
J. G. Amar and F. Family,
Deterministic and stochastic surface growth with generalized nonlinearity, Phys. Rev. E, 47 (1993), 1595-1603.
doi: 10.1103/PhysRevE.47.1595. |
[3] |
J. M. Arrieta, R. B. Anibal and Ph. Souplet,
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 1-15.
|
[4] |
A. Attouchi,
Boundedness of global solutions of a $p$-Laplacian evolution equation with a nonlinear gradient term, Asymptot. Anal., 91 (2015), 233-251.
doi: 10.3233/ASY-141263. |
[5] |
A. Attouchi and Ph. Souplet,
Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion, Trans. Amer. Math. Soc., 369 (2017), 935-974.
doi: 10.1090/tran/6684. |
[6] |
J. Bebernes and S. Bricher,
Final time blowup profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal., 23 (1992), 852-869.
doi: 10.1137/0523045. |
[7] |
J. W. Dold, V. A. Galaktionov, A. A. Lacey and J. L. Vázquez,
Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 663-687.
|
[8] |
M. Fila and G. M. Lieberman,
Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7 (1994), 811-821.
|
[9] |
M. Fila, J. Taskinen and M. Winkler,
Convergence to a singular steady state of a parabolic equation with gradient blow-up, Appl. Math. Lett., 20 (2007), 578-582.
doi: 10.1016/j.aml.2006.07.004. |
[10] |
S. Filippas and R. V. Kohn,
Refined asymptotics for the blow-up of $u_t = \Delta u+u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.
doi: 10.1002/cpa.3160450703. |
[11] |
V. A. Galaktionova and J. R. King,
Fast diffusion equation with critical Sobolev exponent in a ball, Nonlinearity, 15 (2002), 173-188.
doi: 10.1088/0951-7715/15/1/308. |
[12] |
V. A. Galaktionova and J. R. King,
Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differential Equations, 189 (2003), 199-233.
doi: 10.1016/S0022-0396(02)00151-1. |
[13] |
V. A. Galaktionova and J. R. King,
Stabilization to a singular steady state for the Frank-Kamenetskii equation in a critical dimension, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 777-787.
doi: 10.1017/S0308210505000399. |
[14] |
T. Ghoul, V. T. Nguyen and H. Zaag,
Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term, J. Differential Equations, 263 (2017), 4517-4564.
doi: 10.1016/j.jde.2017.05.023. |
[15] |
Y. Giga and R. V. Kohn,
Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.
doi: 10.1002/cpa.3160380304. |
[16] |
Y. J. Guo,
Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Differential Equations, 245 (2008), 809-844.
doi: 10.1016/j.jde.2008.03.012. |
[17] |
J. S. Guo and B. Hu,
Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Syst., 20 (2008), 927-937.
doi: 10.3934/dcds.2008.20.927. |
[18] |
M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez,
Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations, 17 (1992), 205-219.
doi: 10.1080/03605309208820839. |
[19] |
M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez,
Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997.
|
[20] |
M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez,
Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381-450.
|
[21] |
M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 10 (1993), 131–189.
doi: 10.1016/S0294-1449(16)30217-7. |
[22] |
B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-18460-4. |
[23] |
Q. C. Ju, H. L. Li, Y. Li and S. Jiang,
Quasi-neutral limit of the two-fluid Euler-Poisson system, Commun. Pure Appl. Anal., 9 (2010), 1577-1590.
doi: 10.3934/cpaa.2010.9.1577. |
[24] |
M. Kardar, G. Parisi and Y. C. Zhang,
Dynamic scaling of frowing interfaces, Phys. Rev. Lett., 56 (1986), 889-892.
doi: 10.1103/PhysRevLett.56.889. |
[25] |
N. I. Kavallaris and Ph. Souplet,
Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal., 40 (2008), 1852-1881.
doi: 10.1137/080722229. |
[26] |
J. Krug and H. Spohn,
Universality classes for deterministic surface growth, Phys. Rev. A, 38 (1988), 4271-4283.
doi: 10.1103/PhysRevA.38.4271. |
[27] |
Y. X. Li,
Stabilization towards the steady state for a viscous Hamilton-Jacobi equation, Commun. Pure Appl. Anal., 8 (2009), 1917-1924.
doi: 10.3934/cpaa.2009.8.1917. |
[28] |
Y. X. Li and Ph. Souplet,
Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains, Comm. Math. Phys., 293 (2010), 499-517.
doi: 10.1007/s00220-009-0936-8. |
[29] |
G. M. Lieberman,
The first initial-boundary value problem for quasilinear second order parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1986), 347-387.
|
[30] |
Y. Y. Liu, Z. C. Zhang and L. P. Zhu,
Global existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorption, Adv. Differential Equations, 24 (2019), 229-256.
|
[31] |
F. Merle and H. Zaag,
Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., 51 (1998), 139-196.
doi: 10.1002/(SICI)1097-0312(199802)51:2<139::AID-CPA2>3.0.CO;2-C. |
[32] |
Z. Neufeld, M. Vicsek and T. Vicsek,
Complex spatiotemporal patterns in two lattice models with instability, Phys. A, 233 (1996), 754-766.
doi: 10.1016/S0378-4371(96)00188-4. |
[33] |
L. E. Payne and J. C. Song,
Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl., 354 (2009), 394-396.
doi: 10.1016/j.jmaa.2009.01.010. |
[34] |
A. Porretta and Ph. Souplet, The profile of boundary gradient blowup for the diffusive Hamilton-Jacobi equation, Int. Math. Res. Not. IMRN, (2017), No. 17, 5260–5301.
doi: 10.1093/imrn/rnw154. |
[35] |
T. Senba,
Blowup in infinite time of radial solutions for a parabolic-elliptic system in high-dimensional Euclidean spaces, Nonlinear Anal., 70 (2009), 2549-2562.
doi: 10.1016/j.na.2008.03.041. |
[36] |
Ph. Souplet,
Finite time blow-up for a non-linear parabolic equation with a gradient term and applications, Math. Methods Appl. Sci., 19 (1996), 1317-1333.
doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M. |
[37] |
Ph. Souplet,
Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15 (2002), 237-256.
|
[38] |
Ph. Souplet and S. Tayachi,
Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities, Colloq. Math., 88 (2001), 135-154.
doi: 10.4064/cm88-1-10. |
[39] |
Ph. Souplet and J. L. Vázquez,
Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem, Discrete Contin. Dyn. Syst., 14 (2006), 221-234.
doi: 10.3934/dcds.2006.14.221. |
[40] |
Ph. Souplet and Q. S. Zhang,
Global solutions of inhomogeneous Hamilton-Jacobi equations, J. Anal. Math., 99 (2006), 355-396.
doi: 10.1007/BF02789452. |
[41] |
T. I. Zelenyak,
Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differencial'nye Uravnenija, 4 (1968), 34-45.
|
[42] |
Z. C. Zhang and B. Hu,
Gradient blowup rate for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 26 (2010), 767-779.
doi: 10.3934/dcds.2010.26.767. |
[43] |
Z. C. Zhang and B. Hu,
Rate estimates of gradient blowup for a heat equation with exponential nonlinearity, Nonlinear Anal., 72 (2010), 4594-4601.
doi: 10.1016/j.na.2010.02.036. |
[44] |
Z. C. Zhang and Y. Li,
Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3019-3029.
doi: 10.3934/dcdsb.2014.19.3019. |
[45] |
Z. C. Zhang and Y. Li,
Classification of blowup solutions for a parabolic $p$-Laplacian equation with nonlinear gradient terms, J. Math. Anal. Appl., 436 (2016), 1266-1283.
doi: 10.1016/j.jmaa.2015.12.044. |
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and C. P. Grant,
Blow up for a diffusion-advection equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 181-190.
doi: 10.1017/S0308210500024057. |
[2] |
J. G. Amar and F. Family,
Deterministic and stochastic surface growth with generalized nonlinearity, Phys. Rev. E, 47 (1993), 1595-1603.
doi: 10.1103/PhysRevE.47.1595. |
[3] |
J. M. Arrieta, R. B. Anibal and Ph. Souplet,
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 1-15.
|
[4] |
A. Attouchi,
Boundedness of global solutions of a $p$-Laplacian evolution equation with a nonlinear gradient term, Asymptot. Anal., 91 (2015), 233-251.
doi: 10.3233/ASY-141263. |
[5] |
A. Attouchi and Ph. Souplet,
Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion, Trans. Amer. Math. Soc., 369 (2017), 935-974.
doi: 10.1090/tran/6684. |
[6] |
J. Bebernes and S. Bricher,
Final time blowup profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal., 23 (1992), 852-869.
doi: 10.1137/0523045. |
[7] |
J. W. Dold, V. A. Galaktionov, A. A. Lacey and J. L. Vázquez,
Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 663-687.
|
[8] |
M. Fila and G. M. Lieberman,
Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7 (1994), 811-821.
|
[9] |
M. Fila, J. Taskinen and M. Winkler,
Convergence to a singular steady state of a parabolic equation with gradient blow-up, Appl. Math. Lett., 20 (2007), 578-582.
doi: 10.1016/j.aml.2006.07.004. |
[10] |
S. Filippas and R. V. Kohn,
Refined asymptotics for the blow-up of $u_t = \Delta u+u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.
doi: 10.1002/cpa.3160450703. |
[11] |
V. A. Galaktionova and J. R. King,
Fast diffusion equation with critical Sobolev exponent in a ball, Nonlinearity, 15 (2002), 173-188.
doi: 10.1088/0951-7715/15/1/308. |
[12] |
V. A. Galaktionova and J. R. King,
Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differential Equations, 189 (2003), 199-233.
doi: 10.1016/S0022-0396(02)00151-1. |
[13] |
V. A. Galaktionova and J. R. King,
Stabilization to a singular steady state for the Frank-Kamenetskii equation in a critical dimension, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 777-787.
doi: 10.1017/S0308210505000399. |
[14] |
T. Ghoul, V. T. Nguyen and H. Zaag,
Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term, J. Differential Equations, 263 (2017), 4517-4564.
doi: 10.1016/j.jde.2017.05.023. |
[15] |
Y. Giga and R. V. Kohn,
Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.
doi: 10.1002/cpa.3160380304. |
[16] |
Y. J. Guo,
Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Differential Equations, 245 (2008), 809-844.
doi: 10.1016/j.jde.2008.03.012. |
[17] |
J. S. Guo and B. Hu,
Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Syst., 20 (2008), 927-937.
doi: 10.3934/dcds.2008.20.927. |
[18] |
M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez,
Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations, 17 (1992), 205-219.
doi: 10.1080/03605309208820839. |
[19] |
M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez,
Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997.
|
[20] |
M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez,
Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381-450.
|
[21] |
M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 10 (1993), 131–189.
doi: 10.1016/S0294-1449(16)30217-7. |
[22] |
B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-18460-4. |
[23] |
Q. C. Ju, H. L. Li, Y. Li and S. Jiang,
Quasi-neutral limit of the two-fluid Euler-Poisson system, Commun. Pure Appl. Anal., 9 (2010), 1577-1590.
doi: 10.3934/cpaa.2010.9.1577. |
[24] |
M. Kardar, G. Parisi and Y. C. Zhang,
Dynamic scaling of frowing interfaces, Phys. Rev. Lett., 56 (1986), 889-892.
doi: 10.1103/PhysRevLett.56.889. |
[25] |
N. I. Kavallaris and Ph. Souplet,
Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal., 40 (2008), 1852-1881.
doi: 10.1137/080722229. |
[26] |
J. Krug and H. Spohn,
Universality classes for deterministic surface growth, Phys. Rev. A, 38 (1988), 4271-4283.
doi: 10.1103/PhysRevA.38.4271. |
[27] |
Y. X. Li,
Stabilization towards the steady state for a viscous Hamilton-Jacobi equation, Commun. Pure Appl. Anal., 8 (2009), 1917-1924.
doi: 10.3934/cpaa.2009.8.1917. |
[28] |
Y. X. Li and Ph. Souplet,
Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains, Comm. Math. Phys., 293 (2010), 499-517.
doi: 10.1007/s00220-009-0936-8. |
[29] |
G. M. Lieberman,
The first initial-boundary value problem for quasilinear second order parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1986), 347-387.
|
[30] |
Y. Y. Liu, Z. C. Zhang and L. P. Zhu,
Global existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorption, Adv. Differential Equations, 24 (2019), 229-256.
|
[31] |
F. Merle and H. Zaag,
Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., 51 (1998), 139-196.
doi: 10.1002/(SICI)1097-0312(199802)51:2<139::AID-CPA2>3.0.CO;2-C. |
[32] |
Z. Neufeld, M. Vicsek and T. Vicsek,
Complex spatiotemporal patterns in two lattice models with instability, Phys. A, 233 (1996), 754-766.
doi: 10.1016/S0378-4371(96)00188-4. |
[33] |
L. E. Payne and J. C. Song,
Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl., 354 (2009), 394-396.
doi: 10.1016/j.jmaa.2009.01.010. |
[34] |
A. Porretta and Ph. Souplet, The profile of boundary gradient blowup for the diffusive Hamilton-Jacobi equation, Int. Math. Res. Not. IMRN, (2017), No. 17, 5260–5301.
doi: 10.1093/imrn/rnw154. |
[35] |
T. Senba,
Blowup in infinite time of radial solutions for a parabolic-elliptic system in high-dimensional Euclidean spaces, Nonlinear Anal., 70 (2009), 2549-2562.
doi: 10.1016/j.na.2008.03.041. |
[36] |
Ph. Souplet,
Finite time blow-up for a non-linear parabolic equation with a gradient term and applications, Math. Methods Appl. Sci., 19 (1996), 1317-1333.
doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M. |
[37] |
Ph. Souplet,
Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15 (2002), 237-256.
|
[38] |
Ph. Souplet and S. Tayachi,
Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities, Colloq. Math., 88 (2001), 135-154.
doi: 10.4064/cm88-1-10. |
[39] |
Ph. Souplet and J. L. Vázquez,
Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem, Discrete Contin. Dyn. Syst., 14 (2006), 221-234.
doi: 10.3934/dcds.2006.14.221. |
[40] |
Ph. Souplet and Q. S. Zhang,
Global solutions of inhomogeneous Hamilton-Jacobi equations, J. Anal. Math., 99 (2006), 355-396.
doi: 10.1007/BF02789452. |
[41] |
T. I. Zelenyak,
Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differencial'nye Uravnenija, 4 (1968), 34-45.
|
[42] |
Z. C. Zhang and B. Hu,
Gradient blowup rate for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 26 (2010), 767-779.
doi: 10.3934/dcds.2010.26.767. |
[43] |
Z. C. Zhang and B. Hu,
Rate estimates of gradient blowup for a heat equation with exponential nonlinearity, Nonlinear Anal., 72 (2010), 4594-4601.
doi: 10.1016/j.na.2010.02.036. |
[44] |
Z. C. Zhang and Y. Li,
Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3019-3029.
doi: 10.3934/dcdsb.2014.19.3019. |
[45] |
Z. C. Zhang and Y. Li,
Classification of blowup solutions for a parabolic $p$-Laplacian equation with nonlinear gradient terms, J. Math. Anal. Appl., 436 (2016), 1266-1283.
doi: 10.1016/j.jmaa.2015.12.044. |
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