American Institute of Mathematical Sciences

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

Nonexistence results for a fully nonlinear evolution inequality

Qianzhong Ou 1,

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.
Keywords: Liouville theorem, integration by parts., Evolution inequality.
Mathematics Subject Classification: 35K55, 35R4.
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.  doi: 10.1007/BF02392544.  Google Scholar

[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, 13 (1966), 109.   Google Scholar

[3]

K. Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad., 49 (1973), 503.  doi: 10.3792/pja/1195519254.  Google Scholar

[4]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.  doi: 10.1215/S0012-7094-02-11111-9.  Google Scholar

[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.  doi: 10.1007/s00032-004-0032-7.  Google Scholar

[6]

Q. Ou, Nonexistence results for Hessian inequality,, Methods Appl. Anal., 17 (2010), 213.  doi: 10.4310/MAA.2010.v17.n2.a5.  Google Scholar

[7]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.  doi: 10.4007/annals.2008.168.859.  Google Scholar

[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.  doi: 10.1080/03605300600783549.  Google Scholar

[9]

N. Trudinger and X.-J. Wang, Hessian measures. I. Dedicated to Olga Ladyzhenskaya,, Topo. Methods Nonlinear Anal., 10 (1997), 225.   Google Scholar

[10]

N. Trudinger and X.-J. Wang, Hessian measures. II,, Ann. of Math. (2), 150 (1999), 579.  doi: 10.2307/121089.  Google Scholar

[11]

N. Trudinger and X.-J. Wang, Hessian measures. III,, J. Funct. Anal., 193 (2002), 1.  doi: 10.1006/jfan.2001.3925.  Google Scholar

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.  doi: 10.1007/BF02392544.  Google Scholar

[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, 13 (1966), 109.   Google Scholar

[3]

K. Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad., 49 (1973), 503.  doi: 10.3792/pja/1195519254.  Google Scholar

[4]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.  doi: 10.1215/S0012-7094-02-11111-9.  Google Scholar

[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.  doi: 10.1007/s00032-004-0032-7.  Google Scholar

[6]

Q. Ou, Nonexistence results for Hessian inequality,, Methods Appl. Anal., 17 (2010), 213.  doi: 10.4310/MAA.2010.v17.n2.a5.  Google Scholar

[7]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.  doi: 10.4007/annals.2008.168.859.  Google Scholar

[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.  doi: 10.1080/03605300600783549.  Google Scholar

[9]

N. Trudinger and X.-J. Wang, Hessian measures. I. Dedicated to Olga Ladyzhenskaya,, Topo. Methods Nonlinear Anal., 10 (1997), 225.   Google Scholar

[10]

N. Trudinger and X.-J. Wang, Hessian measures. II,, Ann. of Math. (2), 150 (1999), 579.  doi: 10.2307/121089.  Google Scholar

[11]

N. Trudinger and X.-J. Wang, Hessian measures. III,, J. Funct. Anal., 193 (2002), 1.  doi: 10.1006/jfan.2001.3925.  Google Scholar

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