# American Institute of Mathematical Sciences

2016, 23: 19-24. doi: 10.3934/era.2016.23.003

## Nonexistence results for a fully nonlinear evolution inequality

 1 School of Science, Hezhou University, Hezhou, 542899, Guangxi Province, China

Received  January 2015 Revised  April 2016 Published  June 2016

In this paper, a Liouville type theorem is proved for some global fully nonlinear evolution inequality via suitable choices of test functions and the argument of integration by parts.
Citation: Qianzhong Ou. Nonexistence results for a fully nonlinear evolution inequality. Electronic Research Announcements, 2016, 23: 19-24. doi: 10.3934/era.2016.23.003
##### References:
 [1] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301. doi: 10.1007/BF02392544. [2] H. Fujita, On the blowing up of solutions of the Cauchy problems for $u_t=\Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124. [3] K. Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254. [4] D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9. [5] E. Mitidieri and S. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129-162. doi: 10.1007/s00032-004-0032-7. [6] Q. Ou, Nonexistence results for Hessian inequality, Methods Appl. Anal., 17 (2010), 213-223. doi: 10.4310/MAA.2010.v17.n2.a5. [7] N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859. [8] N. C. Phuc and I. E. Verbitsky, Local integral estimates and removable singularities for quasilinear and Hessian equations with nonlinear source terms, Comm. Partial Differential Equations, 31 (2006), 1779-1791. doi: 10.1080/03605300600783549. [9] N. Trudinger and X.-J. Wang, Hessian measures. I. Dedicated to Olga Ladyzhenskaya, Topo. Methods Nonlinear Anal., 10 (1997), 225-239. [10] N. Trudinger and X.-J. Wang, Hessian measures. II, Ann. of Math. (2), 150 (1999), 579-604. doi: 10.2307/121089. [11] N. Trudinger and X.-J. Wang, Hessian measures. III, J. Funct. Anal., 193 (2002), 1-23. doi: 10.1006/jfan.2001.3925.

show all references

##### References:
 [1] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301. doi: 10.1007/BF02392544. [2] H. Fujita, On the blowing up of solutions of the Cauchy problems for $u_t=\Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124. [3] K. Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254. [4] D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9. [5] E. Mitidieri and S. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129-162. doi: 10.1007/s00032-004-0032-7. [6] Q. Ou, Nonexistence results for Hessian inequality, Methods Appl. Anal., 17 (2010), 213-223. doi: 10.4310/MAA.2010.v17.n2.a5. [7] N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859. [8] N. C. Phuc and I. E. Verbitsky, Local integral estimates and removable singularities for quasilinear and Hessian equations with nonlinear source terms, Comm. Partial Differential Equations, 31 (2006), 1779-1791. doi: 10.1080/03605300600783549. [9] N. Trudinger and X.-J. Wang, Hessian measures. I. Dedicated to Olga Ladyzhenskaya, Topo. Methods Nonlinear Anal., 10 (1997), 225-239. [10] N. Trudinger and X.-J. Wang, Hessian measures. II, Ann. of Math. (2), 150 (1999), 579-604. doi: 10.2307/121089. [11] N. Trudinger and X.-J. Wang, Hessian measures. III, J. Funct. Anal., 193 (2002), 1-23. doi: 10.1006/jfan.2001.3925.
 [1] William Guo. Streamlining applications of integration by parts in teaching applied calculus. STEM Education, 2022, 2 (1) : 73-83. doi: 10.3934/steme.2022005 [2] Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131 [3] Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020 [4] Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 [5] Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 [6] Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058 [7] Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155 [8] Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236 [9] Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035 [10] Xian-gao Liu, Xiaotao Zhang. Liouville theorem for MHD system and its applications. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2329-2350. doi: 10.3934/cpaa.2018111 [11] Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549 [12] Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887 [13] Yuan Li. Extremal solution and Liouville theorem for anisotropic elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4063-4082. doi: 10.3934/cpaa.2021144 [14] Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947 [15] Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113 [16] Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511 [17] Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865 [18] Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317 [19] Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248 [20] Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565

2020 Impact Factor: 0.929