Hua CHEN , Huiyang XU . GLOBAL EXISTENCE, EXPONENTIAL DECAY AND BLOW-UP IN FINITE TIME FOR A CLASS OF FINITELY DEGENERATE SEMILINEAR PARABOLIC EQUATIONS[J]. Acta mathematica scientia, Series B, 2019 , 39(5) : 1290 -1308 . DOI: 10.1007/s10473-019-0508-8

[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 Pu_t=-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