[1] Ball J M. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart J Math Oxford Ser, 1977, 28:473-486
[2] Bony J M. Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann Inst Fourier, 1969, 19:277-304
[3] Bramanti M. An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields. Springer Briefs in Mathematics. Springer, 2014
[4] Capogna L, Danielli D, Garofalo N. An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Comm Partial Differential Equations, 1993, 18:1765-1794
[5] Chen H, Liu G W. Global existence and nonexistence for semilinear parabolic equations with conical degeneration. J Pseudo-Differ Oper Appl, 2012, 3:329-349
[6] Chen H, Liu N. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete Contin Dyn Syst, 2016, 36:661-682
[7] Chen H, Luo P, Liu G W. Global solution and blow up of a semilinear heat equation with logarithmic nonlinearity. J Math Anal Appl, 2015, 422:84-98
[8] Chen H, Luo P. Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators. Calc Var Partial Differ Equ, 2015, 54:2831-2852
[9] Di H F, Shang Y D, Zheng X X. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete Contin Dyn Syst Ser B, 2016, 21:781-801
[10] Evans L C. Partial Differential Equations. Amer Math Soc, 2015
[11] Fefferman C, Phong D. Subelliptic eigenvalue problems//Proceedings of the Conference on Harmonic Analysis in Honor of Antoni Zygmund. Wadsworth Math Series, 1981:590-606
[12] Ghezzi R, Jean F. Hausdorff volume in non equiregular sub-Riemannian manifolds. Nonlinear Anal, 2015, 126:345-377
[13] Hajlasz P, Koskela P. Sobolev met Poincaré. Mem Amer Math Soc, 1998, 145
[14] Hörmander L. Hypoelliptic second order differential equations. Acta Math, 1967, 119:147-171
[15] Ioku N. The Cauchy problem for heat equations with exponential nonlinearity. J Differ Equ, 2011, 251:1172-1194
[16] Jerison D. The Poincaré inequality for vector fields satisfying Hörmander's condition. Duke Math J, 1986, 53:503-523
[17] Jerison D, Sánchez-Calle A. Subelliptic, second order differential operators//Complex Analysis III. Lecture Notes in Mathematics 1277. Springer, 1987:46-77
[18] Kohn J J. Hypoellipticity of some degenerate subelliptic operators. J Funct Anal, 1998, 159:203-216
[19] Kohn J J. Subellipticity of the ∂-Neumann problem on pseudo-convex domains:sufficient conditions. Acta Math, 1979, 142:79-122
[20] Komornik V. Exact controllability and stabilization:the multiplier method. SIAM Review, 1994
[21] Levine H A. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=-Au + F(u). Arch Ration Mech Anal, 1973, 51:371-386
[22] Lewy H. An example of a smooth linear partial differential equation without solution. Ann Math, 1957, 66:155-158
[23] Li G, Yu J Y, Liu W J. Global existence, exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration. J Pseudo-Differ Oper Appl, 2017, 8:629-660
[24] Liu W J, Zhuang H F. Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms. Nonlinear Differ Equ Appl, 2017, 24:1-35
[25] Liu Y C, Zhao J S. On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal, 2006, 64:2665-2687
[26] Métivier G. Fonction spectrale et valeurs propres d'une classe d'opérateurs non elliptiques. Comm Partial Differential Equations, 1976, 1:467-519
[27] Montgomery R. A Tour of Subriemannian Geometries, Their Geodesics and Applications. Providence RI:Amer Math Soc, 2002
[28] Morbidelli D. Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields. Studia Mathematica, 2000, 139:213-244
[29] Nagel A, Stein E M, Wainger S. Balls and metrics defined by vector fields I, basic properties. Acta Math, 1985, 155:103-147
[30] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Isr J Math, 1975, 22:273-303
[31] Rothschild L P, Stein E M. Hypoelliptic differential operators and nilpotent groups. Acta Math, 1976, 137:247-320
[32] Sattinger D H. On global solution of nonlinear hyperbolic equations. Arch Ration Mech Anal, 1968, 30:148-172
[33] Schuss Z. Theory and Application of Stochastic Differential Equations. New York:Wiley, 1980
[34] Tan Z. Global solution and blow-up of semilinear heat equation with critical Sobolev exponent. Comm Partial Differential Equations, 2001, 26:717-741
[35] Xu C J. Subelliptic variational problems. Bull Soc Math France, 1990, 118:147-169
[36] Xu C J. Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hörmander's condition. Chinese J Contemp Math, 1994, 15:185-192
[37] Xu R Z, Niu Y. Addendum to "Global existence and finite time blow-up for a class of semilinear pseudoparabolic equations"[J Func Anal, 264(12) (2013) 27322763]. J Funct Anal, 2016, 270:4039-4041
[38] Xu R Z, Zhang M Y, Chen S H, Yang Y B, Shen J H. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete Contin Dyn Syst, 2017, 37:5631-5649
[39] Yang Y B, Lian W, Huang S B, Xu R Z. Finite time blow up of solutions for nonlinear wave equation with general nonlinearity for arbitrarily positive initial energy. Acta Math Sci, 2018, 38A(6):1239-1244
[40] Yung P L. A sharp subelliptic Sobolev embedding theorem with weights. Bull London Math Soc, 2015, 47:396-406 |
is a finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.