Liouville theorems for semilinear differential inequalities on sub-Riemannian manifolds

In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of “good” cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.

[1]  Fabrice Baudoin,et al.  Volume and distance comparison theorems for sub-Riemannian manifolds , 2012, Journal of Functional Analysis.

[2]  Yuhua Sun,et al.  Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold , 2020, Calculus of Variations and Partial Differential Equations.

[3]  A. Grigor’yan,et al.  Superlinear elliptic inequalities on manifolds , 2018, Journal of Functional Analysis.

[4]  N. Tokmagambetov,et al.  Nonexistence Results for the Hyperbolic-Type Equations on Graded Lie Groups , 2020, Bulletin of the Malaysian Mathematical Sciences Society.

[5]  Basilis Gidas,et al.  Global and local behavior of positive solutions of nonlinear elliptic equations , 1981 .

[6]  H. Fujita.  On the blowing up of solutions fo the Cauchy problem for u_t=Δu+u^ , 1966 .

[7]  A. Bonfiglioli,et al.  Nonlinear Liouville theorems for some critical problems on H-type groups , 2004 .

[8]  A. Biswas,et al.  Faber-Krahn type inequalities and uniqueness of positive solutions on metric measure spaces , 2018, Journal of Functional Analysis.

[9]  Jingzhi Tie,et al.  CR Sub-Laplacian Comparison and Liouville-Type Theorem in a Complete Noncompact Sasakian Manifold , 2015, The Journal of Geometric Analysis.

[10]  Xiaohui Yu.  Liouville type theorem in the Heisenberg group with general nonlinearity , 2013 .

[11]  Xiping Zhu,et al.  Yau's gradient estimates on Alexandrov spaces , 2010, 1012.4233.

[12]  L. Véron,et al.  Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group , 2000 .

[13]  M. Ikeda,et al.  Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method , 2019, Nonlinear Analysis.

[14]  S. Pokhozhaev.  Critical nonlinearities in partial differential equations , 2013 .

[15]  Hui-Chun Zhang.  A Note on Liouville Type Theorem of Elliptic Inequality Δu + uσ ⩽ 0 on Riemannian Manifolds , 2015 .

[16]  I. Dolcetta,et al.  Liouville theorems for semilinear equations on the Heisenberg group , 1997 .

[17]  Isoperimetric inequality via Lipschitz regularity of Cheeger-harmonic functions , 2014 .

[18]  Fabrice Baudoin,et al.  A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality , 2010, 1007.1600.

[19]  F. Punzo,et al.  Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds , 2015, 1502.07593.

[20]  Shiping Liu.  Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature , 2020 .

[21]  L. Ambrosio,et al.  Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space , 2017 .

[22]  J. Serrin,et al.  Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities , 2002 .

[23]  M. Bardi,et al.  Liouville results for fully nonlinear equations modeled on Hörmander vector fields: I. The Heisenberg group , 2020, Mathematische Annalen.

[24]  E. Stein,et al.  Balls and metrics defined by vector fields I: Basic properties , 1985 .

[25]  B. Hua,et al.  A note on local gradient estimate on Alexandrov spaces , 2013, 1301.4810.

[26]  E. Mitidieri,et al.  Entire solutions of quasilinear elliptic systems on Carnot groups , 2012, 1209.5087.

[27]  A. Mondino,et al.  Structure theory of metric measure spaces with lower Ricci curvature bounds , 2014, Journal of the European Mathematical Society.

[28]  W. Ni,et al.  Global existence, large time behavior and life span of solutions of a semilinear parabolic cauchy problem , 1992 .

[29]  G. Fitzgerald,et al.  'I. , 2019, Australian journal of primary health.

[30]  Shun-Cheng Chang,et al.  On CR volume growth estimate in a complete pseudohermitian 3-manifold , 2014 .

[31]  Qi S. Zhang.  BLOW-UP RESULTS FOR NONLINEAR PARABOLIC EQUATIONS ON MANIFOLDS , 1999 .

[32]  Fabrice Baudoin,et al.  Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries , 2011, 1101.3590.

[33]  E. Lanconelli,et al.  Existence and nonexistence results for semilinear equations on the Heisenberg group , 1992 .

[34]  B. Yordanov,et al.  Critical Exponent for a Nonlinear Wave Equation with Damping , 2000 .

[35]  A. Palmieri,et al.  Upper bound estimates for local in time solutions to the semilinear heat equation on stratified lie groups in the sub-Fujita case , 2019, RENEWABLE ENERGY SOURCES AND TECHNOLOGIES.

[36]  Lu Xu.  Semi-linear Liouville theorems in the Heisenberg group via vector field methods , 2009 .

[37]  Adam S. Sikora,et al.  Gradient estimates for heat kernels and harmonic functions , 2017, Journal of Functional Analysis.