# American Institute of Mathematical Sciences

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

 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 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 $: $ u $is global and converges to the unique stationary solution in $ C^1 $norm for subcritical boundary value, and $ u_x $blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments. Citation: Caihong Chang, Qiangchang Ju, Zhengce Zhang. Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5991-6014. doi: 10.3934/dcds.2020256 ##### References:  [1] N. D. Alikakos, P. W. Bates and C. P. Grant, Blow up for a diffusion-advection equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 181-190. doi: 10.1017/S0308210500024057. Google Scholar [2] J. G. Amar and F. Family, Deterministic and stochastic surface growth with generalized nonlinearity, Phys. Rev. E, 47 (1993), 1595-1603. doi: 10.1103/PhysRevE.47.1595. Google Scholar [3] J. M. Arrieta, R. B. Anibal and Ph. Souplet, Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 1-15. Google Scholar [4] A. Attouchi, Boundedness of global solutions of a$p$-Laplacian evolution equation with a nonlinear gradient term, Asymptot. Anal., 91 (2015), 233-251. doi: 10.3233/ASY-141263. Google Scholar [5] A. Attouchi and Ph. Souplet, Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with$p$-Laplacian diffusion, Trans. Amer. Math. Soc., 369 (2017), 935-974. doi: 10.1090/tran/6684. Google Scholar [6] J. Bebernes and S. Bricher, Final time blowup profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal., 23 (1992), 852-869. doi: 10.1137/0523045. Google Scholar [7] J. W. Dold, V. A. Galaktionov, A. A. Lacey and J. L. Vázquez, Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 663-687. Google Scholar [8] M. Fila and G. M. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7 (1994), 811-821. Google Scholar [9] M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady state of a parabolic equation with gradient blow-up, Appl. Math. Lett., 20 (2007), 578-582. doi: 10.1016/j.aml.2006.07.004. Google Scholar [10] S. Filippas and R. V. Kohn, Refined asymptotics for the blow-up of$u_t = \Delta u+u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869. doi: 10.1002/cpa.3160450703. Google Scholar [11] V. A. Galaktionova and J. R. King, Fast diffusion equation with critical Sobolev exponent in a ball, Nonlinearity, 15 (2002), 173-188. doi: 10.1088/0951-7715/15/1/308. Google Scholar [12] V. A. Galaktionova and J. R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differential Equations, 189 (2003), 199-233. doi: 10.1016/S0022-0396(02)00151-1. Google Scholar [13] V. A. Galaktionova and J. R. King, Stabilization to a singular steady state for the Frank-Kamenetskii equation in a critical dimension, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 777-787. doi: 10.1017/S0308210505000399. Google Scholar [14] T. Ghoul, V. T. Nguyen and H. Zaag, Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term, J. Differential Equations, 263 (2017), 4517-4564. doi: 10.1016/j.jde.2017.05.023. Google Scholar [15] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304. Google Scholar [16] Y. J. Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Differential Equations, 245 (2008), 809-844. doi: 10.1016/j.jde.2008.03.012. Google Scholar [17] J. S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Syst., 20 (2008), 927-937. doi: 10.3934/dcds.2008.20.927. Google Scholar [18] M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez, Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations, 17 (1992), 205-219. doi: 10.1080/03605309208820839. Google Scholar [19] M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997. Google Scholar [20] M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez, Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381-450. Google Scholar [21] M. A. Herrero and J. J. L. Vel$\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. Google Scholar [22] B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [37] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15 (2002), 237-256. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [45] Z. C. Zhang and Y. Li, Classification of blowup solutions for a parabolic$p$-Laplacian equation with nonlinear gradient terms, J. Math. Anal. Appl., 436 (2016), 1266-1283. doi: 10.1016/j.jmaa.2015.12.044. Google Scholar show all references ##### References:  [1] N. D. Alikakos, P. W. Bates and C. P. Grant, Blow up for a diffusion-advection equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 181-190. doi: 10.1017/S0308210500024057. Google Scholar [2] J. G. Amar and F. Family, Deterministic and stochastic surface growth with generalized nonlinearity, Phys. Rev. E, 47 (1993), 1595-1603. doi: 10.1103/PhysRevE.47.1595. Google Scholar [3] J. M. Arrieta, R. B. Anibal and Ph. Souplet, Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 1-15. Google Scholar [4] A. Attouchi, Boundedness of global solutions of a$p$-Laplacian evolution equation with a nonlinear gradient term, Asymptot. Anal., 91 (2015), 233-251. doi: 10.3233/ASY-141263. Google Scholar [5] A. Attouchi and Ph. Souplet, Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with$p$-Laplacian diffusion, Trans. Amer. Math. Soc., 369 (2017), 935-974. doi: 10.1090/tran/6684. Google Scholar [6] J. Bebernes and S. Bricher, Final time blowup profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal., 23 (1992), 852-869. doi: 10.1137/0523045. Google Scholar [7] J. W. Dold, V. A. Galaktionov, A. A. Lacey and J. L. Vázquez, Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 663-687. Google Scholar [8] M. Fila and G. M. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7 (1994), 811-821. Google Scholar [9] M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady state of a parabolic equation with gradient blow-up, Appl. Math. Lett., 20 (2007), 578-582. doi: 10.1016/j.aml.2006.07.004. Google Scholar [10] S. Filippas and R. V. Kohn, Refined asymptotics for the blow-up of$u_t = \Delta u+u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869. doi: 10.1002/cpa.3160450703. Google Scholar [11] V. A. Galaktionova and J. R. King, Fast diffusion equation with critical Sobolev exponent in a ball, Nonlinearity, 15 (2002), 173-188. doi: 10.1088/0951-7715/15/1/308. Google Scholar [12] V. A. Galaktionova and J. R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differential Equations, 189 (2003), 199-233. doi: 10.1016/S0022-0396(02)00151-1. Google Scholar [13] V. A. Galaktionova and J. R. King, Stabilization to a singular steady state for the Frank-Kamenetskii equation in a critical dimension, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 777-787. doi: 10.1017/S0308210505000399. Google Scholar [14] T. Ghoul, V. T. Nguyen and H. Zaag, Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term, J. Differential Equations, 263 (2017), 4517-4564. doi: 10.1016/j.jde.2017.05.023. Google Scholar [15] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: 10.1002/cpa.3160380304. Google Scholar [16] Y. J. Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Differential Equations, 245 (2008), 809-844. doi: 10.1016/j.jde.2008.03.012. Google Scholar [17] J. S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Syst., 20 (2008), 927-937. doi: 10.3934/dcds.2008.20.927. Google Scholar [18] M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez, Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations, 17 (1992), 205-219. doi: 10.1080/03605309208820839. Google Scholar [19] M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997. Google Scholar [20] M. A. Herrero and J. J. L. Vel$\mathrm{\acute{a}}$zquez, Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381-450. Google Scholar [21] M. A. Herrero and J. J. L. Vel$\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. Google Scholar [22] B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [37] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15 (2002), 237-256. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar
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