Special Issues

Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition

  • Received: 01 December 2019
  • Primary: 35K58; Secondary: 35K60

  • For a class of semilinear parabolic equations under nonlinear dynamical boundary conditions in a bounded domain, we obtain finite time blow-up solutions when the initial data varies in the phase space $ H_0^1(\Omega) $ at positive initial energy level and get global solutions with the initial data at low and critical energy level. Our main tools are potential well method and concavity method.

    Citation: Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition[J]. Electronic Research Archive, 2020, 28(1): 369-381. doi: 10.3934/era.2020021

    Related Papers:

  • For a class of semilinear parabolic equations under nonlinear dynamical boundary conditions in a bounded domain, we obtain finite time blow-up solutions when the initial data varies in the phase space $ H_0^1(\Omega) $ at positive initial energy level and get global solutions with the initial data at low and critical energy level. Our main tools are potential well method and concavity method.



    加载中


    [1] (1975) Sobolev Spaces.Pure and Applied Mathematics, Vol. 65, Academic Press.
    [2] J. M. Arrieta, P. Quittner and A. Rodríguez-Bernal, Parabolic problems with nonlinear dynamical boundary conditions and singular initial data, Differential Integral Equations, 14 (2001), no. 12, 1487–1510.
    [3] J. von Below and G. Pincet Mailly, Blow up for reaction diffusion equations under dynamical boundary conditions, Comm. Partial Differential Equations, 28 (2003), no 1-2,223–247. doi: 10.1081/PDE-120019380
    [4] M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenian. (N.S.), 60 (1991), no. 1, 35–103.
    [5] A. Constantin and J. Escher, Global solutions for quasilinear parabolic problems, J. Evol. Equ., 2 (2002), no. 1, 97–111. doi: 10.1007/s00028-002-8081-2
    [6] A. Constantin, J. Escher and Z. Yin, Global solutions for quasilinear parabolic systems, J. Differential Equations, 197 (2004), no. 1, 73–84. doi: 10.1016/S0022-0396(03)00165-7
    [7] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), no. 7-8, 1309–1364. doi: 10.1080/03605309308820976
    [8] J. Escher, On quasilinear fully parabolic boundary value problems, Differential Integral Equations, 7 (1994), no. 5-6, 1325–1343.
    [9] J. Escher, On the qualitative behaviour of some semilinear parabolic problems, Differential Integral Equations, 8 (1995), no. 2,247–267.
    [10] Attractors for parabolic equations with dynamic boundary conditions. Nonlinear Anal. (2008) 68: 1723-1732.
    [11] M. Fila and P. Quittner, Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition, in Topics in Nonlinear Analysis, Vol. 35, Birkhäuser, Basel, 1999,251–272. doi: 10.1007/978-3-0348-8765-6_12
    [12] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), no. 2,185–207. doi: 10.1016/j.anihpc.2005.02.007
    [13] T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), no. 1-2, 43–60. doi: 10.1017/S0308210500023945
    [14] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), no. 1,613–632. doi: 10.1515/anona-2020-0016
    [15] Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D., 237 (2008), no. 6,721–731. doi: 10.1016/j.physd.2007.09.028
    [16] Y. Liu, R. Xu and T. Yu, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations, Nonlinear Anal., 68 (2008), no. 11, 3332–3348. doi: 10.1016/j.na.2007.03.029
    [17] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), no. 3-4,273–303. doi: 10.1007/BF02761595
    [18] J. F. Rodrigues, V. A. Solonnikov and F. Yi, On a parabolic system with time derivative in the boundary conditions and related free boundary problems, Math. Ann., 315 (1999), no. 1, 61–95. doi: 10.1007/s002080050318
    [19] E. Vitillaro, Global existence for the heat equation with nonlinear dynamical boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), no. 1,175–207. doi: 10.1017/S0308210500003838
    [20] H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition, Adv. Math. Sci. Appl., 17 (2007), no. 1, 67–88.
    [21] R. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), no. 3,459–468. doi: 10.1090/S0033-569X-2010-01197-0
    [22] R. Xu, W. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), no. 2,321–356. doi: 10.1007/s11425-017-9280-x
    [23] Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. (2013) 264: 2732-2763.
    [24] Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy. Appl. Math. Lett. (2018) 83: 176-181.
    [25] R. Xu, M. Zhang, S. Chen, Y. Yang and J. Shen, The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), no. 11, 5631–5649. doi: 10.3934/dcds.2017244
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1671) PDF downloads(245) Cited by(5)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog