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December  2021, 14(12): 4179-4200. doi: 10.3934/dcdss.2021109

Global existence and nonexistence for a class of finitely degenerate coupled parabolic systems with high initial energy level

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

* Corresponding author: Jiangbo Han

Received  July 2021 Revised  August 2021 Published  December 2021 Early access  October 2021

Fund Project: The first author is supported by the NSFHPC under the grants LH2021A001

In this paper, we consider a class of finitely degenerate coupled parabolic systems. At high initial energy level $ J(u_{0})>d $, we present a new sufficient condition to describe the global existence and nonexistence of solutions for problem (1)-(4) respectively. Moreover, by applying the Levine's concavity method, we give some affirmative answers to finite time blow up of solutions at arbitrary positive initial energy $ J(u_{0})>0 $, including the estimate of upper bound of blowup time.

Citation: Yuxuan Chen, Jiangbo Han. Global existence and nonexistence for a class of finitely degenerate coupled parabolic systems with high initial energy level. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4179-4200. doi: 10.3934/dcdss.2021109
References:
[1]

L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math., 50 (1997), 867-889.  doi: 10.1002/(SICI)1097-0312(199709)50:9<867::AID-CPA3>3.0.CO;2-3.

[2]

L. Capogna, Regularity for quasilinear equations and $1$-quasiconformal maps in Carnot groups, Math. Ann., 313 (1999), 263-295.  doi: 10.1007/s002080050261.

[3]

H. Chen and G. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 3 (2012), 329-349.  doi: 10.1007/s11868-012-0046-9.

[4]

H. Chen and N. Liu, Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials, Discrete Contin. Dyn. Syst., 36 (2016), 661-682.  doi: 10.3934/dcds.2016.36.661.

[5]

H. ChenX. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities, Calc. Var. Partial Differential Equations, 43 (2012), 463-484.  doi: 10.1007/s00526-011-0418-7.

[6]

H. Chen and P. Luo, Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators, Calc. Var. Partial Differential Equations, 54 (2015), 2831-2852.  doi: 10.1007/s00526-015-0885-3.

[7]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.

[8]

H. Chen, J. Wang and H. Y. Xu, Global existence, exponential decay and finite time blow-up for a class of finitely degenerate coupled parabolic systems, Methods Appl. Anal..

[9]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[10]

H. Chen and H. Xu, Global existence, exponential decay and blow-up in finite time for a class of finitely degenerate semilinear parabolic equations, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 1290-1308.  doi: 10.1007/s10473-019-0508-8.

[11]

W. Chen and M. Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $T^n$, Comm. Partial Differential Equations, 25 (2000), 337-354.  doi: 10.1080/03605300008821516.

[12]

K.-S. Chou and Y.-C. Kwong, General initial data for a class of parabolic equations including the curve shortening problem, Discrete Contin. Dyn. Syst., 40 (2020), 2963-2986.  doi: 10.3934/dcds.2020157.

[13]

M. Christ, Hypoellipticity in the infinitely degenerate regime, in Complex Analysis and Geometry (Columbus, OH, 1999), vol. 9 of Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin, (2001), 59–84.

[14]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[15]

H. Ding and J. Zhou, Global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem, Nonlinearity, 33 (2020), 6099-6133.  doi: 10.1088/1361-6544/ab9f84.

[16]

R. DuJ. EichhornQ. Liu and C. Wang, Carleman estimates and null controllability of a class of singular parabolic equations, Adv. Nonlinear Anal., 8 (2019), 1057-1082.  doi: 10.1515/anona-2016-0266.

[17]

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.

[18]

M. Escobedo and H. A. Levine, Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch. Rational Mech. Anal., 129 (1995), 47-100.  doi: 10.1007/BF00375126.

[19]

C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, (1983), 590–606.

[20]

N. Garofalo and D. Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J., 106 (2001), 411-448.  doi: 10.1215/S0012-7094-01-10631-5.

[21]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990. 

[22]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.

[23]

D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J., 53 (1986), 503-523.  doi: 10.1215/S0012-7094-86-05329-9.

[24]

D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, in Microlocal Analysis (Boulder, Colo., 1983), vol. 27 of Contemp. Math., Amer. Math. Soc., Providence, RI, (1984), 57–63. doi: 10.1090/conm/027/741039.

[25]

D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197.  doi: 10.4310/jdg/1214440849.

[26]

D. S. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J., 35 (1986), 835-854.  doi: 10.1512/iumj.1986.35.35043.

[27]

J. J. Kohn, Subellipticity of the $\bar \partial $-Neumann problem on pseudo-convex domains: Sufficient conditions, Acta Math., 142 (1979), 79-122.  doi: 10.1007/BF02395058.

[28]

J. J. Kohn, Hypoellipticity of some degenerate subelliptic operators, J. Funct. Anal., 159 (1998), 203-216.  doi: 10.1006/jfan.1998.3289.

[29]

J. J. Kohn, Hypoellipticity at points of infinite type, in Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998), vol. 251 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2000), 393–398. doi: 10.1090/conm/251/03882.

[30]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_{t}=-Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[31]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[32]

G. Métivier, Fonction spectrale et valeurs propres d'une classe d'opérateurs non elliptiques, Comm. Partial Differential Equations, 1 (1976), 467-519.  doi: 10.1080/03605307608820018.

[33]

M. Nakao, Global existence to the initial-boundary value problem for a system of nonlinear diffusion and wave equations, J. Differential Equations, 264 (2018), 134-162.  doi: 10.1016/j.jde.2017.09.001.

[34]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[35]

T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.

[36]

R. B. Salako, Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 39 (2019), 5945-5973.  doi: 10.3934/dcds.2019260.

[37]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118 (1983), 525-571.  doi: 10.2307/2006981.

[38]

A. SlavíkP. Stehlík and J. Volek, Well-posedness and maximum principles for lattice reaction-diffusion equations, Adv. Nonlinear Anal., 8 (2019), 303-322.  doi: 10.1515/anona-2016-0116.

[39]

M. Sônego, On the weakly degenerate Allen-Cahn equation, Adv. Nonlinear Anal., 9 (2020), 361-371.  doi: 10.1515/anona-2020-0004.

[40]

F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis, 32 (1979), 277-296.  doi: 10.1016/0022-1236(79)90040-5.

[41]

C. J. Xu, Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math., 45 (1992), 77-96.  doi: 10.1002/cpa.3160450104.

[42]

C. J. Xu, Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hörmander's condition, Chinese J. Contemp. Math., 15 (1994), 183-192. 

[43]

C.-J. Xu and C. Zuily, Higher interior regularity for quasilinear subelliptic systems, Calc. Var. Partial Differential Equations, 5 (1997), 323-343.  doi: 10.1007/s005260050069.

[44]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.

[45]

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.

[46]

P.-L. Yung, A sharp subelliptic Sobolev embedding theorem with weights, Bull. Lond. Math. Soc., 47 (2015), 396-406.  doi: 10.1112/blms/bdv010.

[47]

M. ZhangQ. ZhaoY. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.

[48]

Y. Zhang and M. Feng, A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior, Electron. Res. Arch., 28 (2020), 1419-1438.  doi: 10.3934/era.2020075.

[49]

J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.

show all references

References:
[1]

L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math., 50 (1997), 867-889.  doi: 10.1002/(SICI)1097-0312(199709)50:9<867::AID-CPA3>3.0.CO;2-3.

[2]

L. Capogna, Regularity for quasilinear equations and $1$-quasiconformal maps in Carnot groups, Math. Ann., 313 (1999), 263-295.  doi: 10.1007/s002080050261.

[3]

H. Chen and G. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 3 (2012), 329-349.  doi: 10.1007/s11868-012-0046-9.

[4]

H. Chen and N. Liu, Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials, Discrete Contin. Dyn. Syst., 36 (2016), 661-682.  doi: 10.3934/dcds.2016.36.661.

[5]

H. ChenX. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities, Calc. Var. Partial Differential Equations, 43 (2012), 463-484.  doi: 10.1007/s00526-011-0418-7.

[6]

H. Chen and P. Luo, Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators, Calc. Var. Partial Differential Equations, 54 (2015), 2831-2852.  doi: 10.1007/s00526-015-0885-3.

[7]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.

[8]

H. Chen, J. Wang and H. Y. Xu, Global existence, exponential decay and finite time blow-up for a class of finitely degenerate coupled parabolic systems, Methods Appl. Anal..

[9]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[10]

H. Chen and H. Xu, Global existence, exponential decay and blow-up in finite time for a class of finitely degenerate semilinear parabolic equations, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 1290-1308.  doi: 10.1007/s10473-019-0508-8.

[11]

W. Chen and M. Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $T^n$, Comm. Partial Differential Equations, 25 (2000), 337-354.  doi: 10.1080/03605300008821516.

[12]

K.-S. Chou and Y.-C. Kwong, General initial data for a class of parabolic equations including the curve shortening problem, Discrete Contin. Dyn. Syst., 40 (2020), 2963-2986.  doi: 10.3934/dcds.2020157.

[13]

M. Christ, Hypoellipticity in the infinitely degenerate regime, in Complex Analysis and Geometry (Columbus, OH, 1999), vol. 9 of Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin, (2001), 59–84.

[14]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[15]

H. Ding and J. Zhou, Global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem, Nonlinearity, 33 (2020), 6099-6133.  doi: 10.1088/1361-6544/ab9f84.

[16]

R. DuJ. EichhornQ. Liu and C. Wang, Carleman estimates and null controllability of a class of singular parabolic equations, Adv. Nonlinear Anal., 8 (2019), 1057-1082.  doi: 10.1515/anona-2016-0266.

[17]

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.

[18]

M. Escobedo and H. A. Levine, Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch. Rational Mech. Anal., 129 (1995), 47-100.  doi: 10.1007/BF00375126.

[19]

C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, (1983), 590–606.

[20]

N. Garofalo and D. Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J., 106 (2001), 411-448.  doi: 10.1215/S0012-7094-01-10631-5.

[21]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990. 

[22]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.

[23]

D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J., 53 (1986), 503-523.  doi: 10.1215/S0012-7094-86-05329-9.

[24]

D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, in Microlocal Analysis (Boulder, Colo., 1983), vol. 27 of Contemp. Math., Amer. Math. Soc., Providence, RI, (1984), 57–63. doi: 10.1090/conm/027/741039.

[25]

D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197.  doi: 10.4310/jdg/1214440849.

[26]

D. S. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J., 35 (1986), 835-854.  doi: 10.1512/iumj.1986.35.35043.

[27]

J. J. Kohn, Subellipticity of the $\bar \partial $-Neumann problem on pseudo-convex domains: Sufficient conditions, Acta Math., 142 (1979), 79-122.  doi: 10.1007/BF02395058.

[28]

J. J. Kohn, Hypoellipticity of some degenerate subelliptic operators, J. Funct. Anal., 159 (1998), 203-216.  doi: 10.1006/jfan.1998.3289.

[29]

J. J. Kohn, Hypoellipticity at points of infinite type, in Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998), vol. 251 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2000), 393–398. doi: 10.1090/conm/251/03882.

[30]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_{t}=-Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[31]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[32]

G. Métivier, Fonction spectrale et valeurs propres d'une classe d'opérateurs non elliptiques, Comm. Partial Differential Equations, 1 (1976), 467-519.  doi: 10.1080/03605307608820018.

[33]

M. Nakao, Global existence to the initial-boundary value problem for a system of nonlinear diffusion and wave equations, J. Differential Equations, 264 (2018), 134-162.  doi: 10.1016/j.jde.2017.09.001.

[34]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[35]

T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.

[36]

R. B. Salako, Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 39 (2019), 5945-5973.  doi: 10.3934/dcds.2019260.

[37]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118 (1983), 525-571.  doi: 10.2307/2006981.

[38]

A. SlavíkP. Stehlík and J. Volek, Well-posedness and maximum principles for lattice reaction-diffusion equations, Adv. Nonlinear Anal., 8 (2019), 303-322.  doi: 10.1515/anona-2016-0116.

[39]

M. Sônego, On the weakly degenerate Allen-Cahn equation, Adv. Nonlinear Anal., 9 (2020), 361-371.  doi: 10.1515/anona-2020-0004.

[40]

F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis, 32 (1979), 277-296.  doi: 10.1016/0022-1236(79)90040-5.

[41]

C. J. Xu, Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math., 45 (1992), 77-96.  doi: 10.1002/cpa.3160450104.

[42]

C. J. Xu, Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hörmander's condition, Chinese J. Contemp. Math., 15 (1994), 183-192. 

[43]

C.-J. Xu and C. Zuily, Higher interior regularity for quasilinear subelliptic systems, Calc. Var. Partial Differential Equations, 5 (1997), 323-343.  doi: 10.1007/s005260050069.

[44]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.

[45]

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.

[46]

P.-L. Yung, A sharp subelliptic Sobolev embedding theorem with weights, Bull. Lond. Math. Soc., 47 (2015), 396-406.  doi: 10.1112/blms/bdv010.

[47]

M. ZhangQ. ZhaoY. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.

[48]

Y. Zhang and M. Feng, A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior, Electron. Res. Arch., 28 (2020), 1419-1438.  doi: 10.3934/era.2020075.

[49]

J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.

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