American Institute of Mathematical Sciences

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March  2020, 28(1): 369-381. doi: 10.3934/era.2020021

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

Mingyou Zhang 1, , Qingsong Zhao 2, , Yu Liu 3, and  Wenke Li 4,,

1. 

College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China
2. 

School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430000, China
3. 

College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang 150001, China
4. 

College of Power and Energy Engineering, Harbin Engineering University, Harbin, Heilongjiang 150001, China

* Corresponding author: Wenke Li, liwenke@hrbeu.edu.cn

Received  December 2019 Published  March 2020

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.

Keywords: Global existence, blow up, nonlinear dynamical boundary, potential well method, concavity method.
Mathematics Subject Classification: Primary: 35K58; Secondary: 35K60.
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. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
References:
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[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), no. 7-8, 1309–1364. doi: 10.1080/03605309308820976.  Google Scholar

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Z.-H. Fan and C.-K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal., 68 (2008), 1723-1732.  doi: 10.1016/j.na.2007.01.005.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[24]

R. Xu, X. Wang and Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Escher, On quasilinear fully parabolic boundary value problems, Differential Integral Equations, 7 (1994), no. 5-6, 1325–1343.  Google Scholar

[9]

J. Escher, On the qualitative behaviour of some semilinear parabolic problems, Differential Integral Equations, 8 (1995), no. 2,247–267.  Google Scholar

[10]

Z.-H. Fan and C.-K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal., 68 (2008), 1723-1732.  doi: 10.1016/j.na.2007.01.005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[24]

R. Xu, X. Wang and Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[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.  Google Scholar

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