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

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.

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:
[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. XuX. 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. XuX. 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

[1]

Walter A. Strauss, Kimitoshi Tsutaya. Existence and blow up of small amplitude nonlinear waves with a negative potential. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 175-188. doi: 10.3934/dcds.1997.3.175

[2]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621

[3]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042

[4]

Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63

[5]

Joachim von Below, Gaëlle Pincet Mailly. Blow up for some nonlinear parabolic problems with convection under dynamical boundary conditions. Conference Publications, 2007, 2007 (Special) : 1031-1041. doi: 10.3934/proc.2007.2007.1031

[6]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[7]

Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015

[8]

Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721

[9]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535

[10]

Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827

[11]

Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043

[12]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169

[13]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[14]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[15]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[16]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[17]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671

[18]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[19]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355

[20]

Joachim von Below, Gaëlle Pincet Mailly, Jean-François Rault. Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 825-836. doi: 10.3934/dcdss.2013.6.825

2018 Impact Factor: 0.263

Article outline

[Back to Top]