doi: 10.3934/dcdss.2021109
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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 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 & Continuous Dynamical Systems - S, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[41]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[41]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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