doi: 10.3934/dcdsb.2022107
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Asymptotic behavior of solutions to coupled semilinear parabolic equations with general degenerate diffusion coefficients

School of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Chunpeng Wang

Received  April 2021 Revised  April 2022 Early access June 2022

Fund Project: Supported by National Key R & D Program of China (No. 2020YFA0714101), and National Natural Science Foundation of China (No. 11925105 and 11871133)

This paper concerns the asymptotic behavior of solutions to one-dimensional coupled semilinear degenerate parabolic equations with superlinear reaction terms both in bounded and unbounded intervals. The equations are degenerate at a lateral boundary point and the diffusion coefficients are general functions. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large ones in the case that the degeneracy is not strong. Whereas in the case that the degeneracy is strong enough, the nontrivial solution must blow up in a finite time. For the problem in an unbounded interval, blowing-up theorems of Fujita type are established. If the degeneracy of the equations at the lateral boundary point is strong enough, it is shown that any nontrivial solution must blow up in a finite time. If the degeneracy of the equations at the lateral boundary point is not strong, it is proved that the critical Fujita curve is determined by the asymptotic behavior of the diffusion coefficient at infinity. Furthermore, the critical case is also considered.

Citation: Xinxin Jing, Yuanyuan Nie, Chunpeng Wang. Asymptotic behavior of solutions to coupled semilinear parabolic equations with general degenerate diffusion coefficients. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022107
References:
[1]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[2]

D. AndreucciG. R. CirmiS. Leonardi and A. F. Tedeev, Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differential Equations, 174 (2001), 253-288.  doi: 10.1006/jdeq.2000.3948.

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.

[4]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), No. 136, 20 pp.

[5]

P. CannarsaG. Fragnelli and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl., 320 (2006), 804-818.  doi: 10.1016/j.jmaa.2005.07.006.

[6]

P. CannarsaP. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal., 3 (2004), 607-635.  doi: 10.3934/cpaa.2004.3.607.

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P. CannarsaP. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190. 

[8]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.

[9]

P. Cannarsa and L. de Teresa, Controllability of 1-D coupled degenerate parabolic equations, Electron. J. Differential Equations, 2009 (2009), No. 73, 21 pp.

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K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl., 243 (2000), 85-126.  doi: 10.1006/jmaa.1999.6663.

[11]

R. DuC. Wang and Q. Zhou, Approximate controllability of a semilinear system involving a fully nonlinear gradient term, Appl. Math. Optim., 70 (2014), 165-183.  doi: 10.1007/s00245-014-9238-4.

[12]

M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202.  doi: 10.1016/0022-0396(91)90118-S.

[13]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. 

[14]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad, 49 (1973), 503-505.  doi: 10.3792/pja/1195519254.

[15]

X. Jing, Y. Nie and C. Wang, Asymptotic behavior of solutions to coupled semilinear parabolic systems with boundary degeneracy, Electron. J. Differential Equations, 2021 (2021), 17 pp.

[16]

X. Jing, C. Wang and M. Zhou, Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy, to appear, Communications in Mathematical Research.

[17]

K. KobayashiT. Siaro and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424.  doi: 10.2969/jmsj/02930407.

[18]

H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32 (1990), 262-288.  doi: 10.1137/1032046.

[19]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362.  doi: 10.1007/s00028-006-0214-6.

[20]

Y. NaY. Nie and X. Zhou, Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms, J. Nonlinear Sci. Appl., 10 (2017), 5813-5824.  doi: 10.22436/jnsa.010.11.19.

[21]

G. R. NorthL. HowardD. Pollard and B. Wielicki, Variational formulation of Budyko-Sellers climate models, J. Atmospheric Sci., 36 (1979), 255-259.  doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2.

[22]

C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy, J. Evol. Equ., 10 (2010), 163-193.  doi: 10.1007/s00028-009-0044-4.

[23]

C. Wang, Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy, Proc. Amer. Math. Soc., 141 (2013), 3125-3140.  doi: 10.1090/S0002-9939-2013-11945-3.

[24]

C. Wang, Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy, Discrete Contin. Dyn. Syst., 36 (2016), 1041-1060.  doi: 10.3934/dcds.2016.36.1041.

[25]

C. Wang and R. Du, Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, SIAM J. Control Optim., 52 (2014), 1457-1480.  doi: 10.1137/110820592.

[26]

C. Wang and R. Du, Approximate controllability of a class of semilinear degenerate systems with convection term, J. Differential Equations, 254 (2013), 3665-3689.  doi: 10.1016/j.jde.2013.01.038.

[27]

C. Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 415-430.  doi: 10.1017/S0308210500004637.

[28]

C. WangS. Zheng and Z. Wang, Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity, 20 (2007), 1343-1359.  doi: 10.1088/0951-7715/20/6/002.

[29]

C. WangY. ZhouR. Du and Q. Liu, Carleman estimate for solutions to a degenerate convection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4207-4222.  doi: 10.3934/dcdsb.2018133.

[30]

J. Xu, C. Wang and Y. Nie, Carleman estimate and null controllability of a cascade degenerate parabolic system with general convection terms, Electron. J. Differential Equations, 2018 (2018), Paper No. 195, 20 pp.

[31]

S. Zheng, Global existence and global non-existence of solutions to a reaction-diffusion system, Nonlinear Anal., 39 (2000), 327-340.  doi: 10.1016/S0362-546X(98)00171-0.

[32]

S. ZhengX. Song and Z. Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl., 298 (2004), 308-324.  doi: 10.1016/j.jmaa.2004.05.017.

[33]

S. Zheng and C. Wang, Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms, Nonlinearity, 21 (2008), 2179-2200.  doi: 10.1088/0951-7715/21/9/015.

[34]

Q. ZhouY. Nie and X. Han, Large time behavior of solutions to semilinear parabolic equations with gradient, J. Dyn. Control Syst., 22 (2016), 191-205.  doi: 10.1007/s10883-015-9294-3.

show all references

References:
[1]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[2]

D. AndreucciG. R. CirmiS. Leonardi and A. F. Tedeev, Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differential Equations, 174 (2001), 253-288.  doi: 10.1006/jdeq.2000.3948.

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.

[4]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), No. 136, 20 pp.

[5]

P. CannarsaG. Fragnelli and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl., 320 (2006), 804-818.  doi: 10.1016/j.jmaa.2005.07.006.

[6]

P. CannarsaP. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal., 3 (2004), 607-635.  doi: 10.3934/cpaa.2004.3.607.

[7]

P. CannarsaP. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190. 

[8]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.

[9]

P. Cannarsa and L. de Teresa, Controllability of 1-D coupled degenerate parabolic equations, Electron. J. Differential Equations, 2009 (2009), No. 73, 21 pp.

[10]

K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl., 243 (2000), 85-126.  doi: 10.1006/jmaa.1999.6663.

[11]

R. DuC. Wang and Q. Zhou, Approximate controllability of a semilinear system involving a fully nonlinear gradient term, Appl. Math. Optim., 70 (2014), 165-183.  doi: 10.1007/s00245-014-9238-4.

[12]

M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202.  doi: 10.1016/0022-0396(91)90118-S.

[13]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. 

[14]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad, 49 (1973), 503-505.  doi: 10.3792/pja/1195519254.

[15]

X. Jing, Y. Nie and C. Wang, Asymptotic behavior of solutions to coupled semilinear parabolic systems with boundary degeneracy, Electron. J. Differential Equations, 2021 (2021), 17 pp.

[16]

X. Jing, C. Wang and M. Zhou, Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy, to appear, Communications in Mathematical Research.

[17]

K. KobayashiT. Siaro and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424.  doi: 10.2969/jmsj/02930407.

[18]

H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32 (1990), 262-288.  doi: 10.1137/1032046.

[19]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362.  doi: 10.1007/s00028-006-0214-6.

[20]

Y. NaY. Nie and X. Zhou, Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms, J. Nonlinear Sci. Appl., 10 (2017), 5813-5824.  doi: 10.22436/jnsa.010.11.19.

[21]

G. R. NorthL. HowardD. Pollard and B. Wielicki, Variational formulation of Budyko-Sellers climate models, J. Atmospheric Sci., 36 (1979), 255-259.  doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2.

[22]

C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy, J. Evol. Equ., 10 (2010), 163-193.  doi: 10.1007/s00028-009-0044-4.

[23]

C. Wang, Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy, Proc. Amer. Math. Soc., 141 (2013), 3125-3140.  doi: 10.1090/S0002-9939-2013-11945-3.

[24]

C. Wang, Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy, Discrete Contin. Dyn. Syst., 36 (2016), 1041-1060.  doi: 10.3934/dcds.2016.36.1041.

[25]

C. Wang and R. Du, Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, SIAM J. Control Optim., 52 (2014), 1457-1480.  doi: 10.1137/110820592.

[26]

C. Wang and R. Du, Approximate controllability of a class of semilinear degenerate systems with convection term, J. Differential Equations, 254 (2013), 3665-3689.  doi: 10.1016/j.jde.2013.01.038.

[27]

C. Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 415-430.  doi: 10.1017/S0308210500004637.

[28]

C. WangS. Zheng and Z. Wang, Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity, 20 (2007), 1343-1359.  doi: 10.1088/0951-7715/20/6/002.

[29]

C. WangY. ZhouR. Du and Q. Liu, Carleman estimate for solutions to a degenerate convection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4207-4222.  doi: 10.3934/dcdsb.2018133.

[30]

J. Xu, C. Wang and Y. Nie, Carleman estimate and null controllability of a cascade degenerate parabolic system with general convection terms, Electron. J. Differential Equations, 2018 (2018), Paper No. 195, 20 pp.

[31]

S. Zheng, Global existence and global non-existence of solutions to a reaction-diffusion system, Nonlinear Anal., 39 (2000), 327-340.  doi: 10.1016/S0362-546X(98)00171-0.

[32]

S. ZhengX. Song and Z. Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl., 298 (2004), 308-324.  doi: 10.1016/j.jmaa.2004.05.017.

[33]

S. Zheng and C. Wang, Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms, Nonlinearity, 21 (2008), 2179-2200.  doi: 10.1088/0951-7715/21/9/015.

[34]

Q. ZhouY. Nie and X. Han, Large time behavior of solutions to semilinear parabolic equations with gradient, J. Dyn. Control Syst., 22 (2016), 191-205.  doi: 10.1007/s10883-015-9294-3.

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