Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

Caihong Chang; Qiangchang Ju; Zhengce Zhang

doi:10.3934/dcds.2020256

Previous Article
Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems
DCDS Home
This Issue
Next Article
Filtering the $ L^2- $critical focusing Schrödinger equation

October 2020, 40(10): 5991-6014. doi: 10.3934/dcds.2020256

Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

Caihong Chang ^1, , Qiangchang Ju ^2, and Zhengce Zhang ^1,,

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

^* Corresponding author: Zhengce Zhang

Received March 2020 Revised May 2020 Published June 2020

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term:

$ u_t = u_{xx}+x^m|u_x|^p, \ \ t>0, \ \ 0<x<1, $

where

$ p>m+2 $

$ m\geq0 $

. Zhang and Hu [Discrete Contin. Dyn. Syst. 26 (2010) 767-779] showed that finite time gradient blowup occurs at the boundary and the accurate blowup rate is also obtained for super-critical boundary value. Throughout this paper, we present a complete large time behavior of a classical solution

$ u $

is global and converges to the unique stationary solution in

$ C^1 $

norm for subcritical boundary value, and

$ u_x $

blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments.

Keywords: Asymptotic behavior, boundedness, gradient nonlinearity, gradient growup rate, Lyapunov functional, matched asymptotic expansions.

Mathematics Subject Classification: 35A01, 35B40, 35B44, 35K20.

Citation: Caihong Chang, Qiangchang Ju, Zhengce Zhang. Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5991-6014. doi: 10.3934/dcds.2020256

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.

[1]	Ignacio Guerra. A semilinear problem with a gradient term in the nonlinearity. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 137-162. doi: 10.3934/dcds.2021110
[2]	Yan Zhang. Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5183-5199. doi: 10.3934/dcds.2016025
[3]	Zhengce Zhang, Bei Hu. Gradient blowup rate for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 767-779. doi: 10.3934/dcds.2010.26.767
[4]	Chao Liu, Bin Liu. Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021255
[5]	Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness and asymptotic behavior of solutions for a quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022118
[6]	Zixiao Liu, Jiguang Bao. Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022081
[7]	Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145
[8]	Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721
[9]	Zongming Guo, Juncheng Wei. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Communications on Pure and Applied Analysis, 2008, 7 (4) : 765-786. doi: 10.3934/cpaa.2008.7.765
[10]	Mamoru Okamoto. Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity. Evolution Equations and Control Theory, 2019, 8 (3) : 567-601. doi: 10.3934/eect.2019027
[11]	Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada. A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2719-2745. doi: 10.3934/dcds.2021209
[12]	Chunyan Zhao, Chengkui Zhong, Zhijun Tang. Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022025
[13]	Feng Li, Erik Lindgren. Large time behavior for a nonlocal nonlinear gradient flow. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022079
[14]	Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1
[15]	Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems and Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041
[16]	Danilo Costarelli, Gianluca Vinti. Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators. Mathematical Foundations of Computing, 2020, 3 (1) : 41-50. doi: 10.3934/mfc.2020004
[17]	Jiahui Yu, Konstantinos Spiliopoulos. Normalization effects on shallow neural networks and related asymptotic expansions. Foundations of Data Science, 2021, 3 (2) : 151-200. doi: 10.3934/fods.2021013
[18]	Jonathan Zinsl. The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 919-933. doi: 10.3934/dcdss.2017047
[19]	Fen-Fen Yang. Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise. Probability, Uncertainty and Quantitative Risk, , () : -. doi: 10.3934/puqr.2022008
[20]	Jagmohan Tyagi, Ram Baran Verma. Positive solution to extremal Pucci's equations with singular and gradient nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2637-2659. doi: 10.3934/dcds.2019110

American Institute of Mathematical Sciences

Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors