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Some nonexistence results for higher-order evolution inequalities in cone-like domains


Author: Gennady G. Laptev
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 87-93
MSC (2000): Primary 35G25; Secondary 35R45, 35K55, 35L70
DOI: https://doi.org/10.1090/S1079-6762-01-00098-1
Published electronically: October 15, 2001
MathSciNet review: 1856890
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Abstract | References | Similar Articles | Additional Information

Abstract: Nonexistence of global (positive) solutions of semilinear higher-order evolution inequalities

\begin{displaymath}\frac{\partial^k u}{\partial t^k}-\Delta u^m\ge \vert u\vert^... ...\partial t^k}-\text{\rm div}\, (\vert x\vert^\alpha Du)\ge u^q \end{displaymath}

with $k=1,2,\dots$, in cone-like domains is studied. The critical exponents $q^*$ are found and the nonexistence results are proved for $1<q\le q^*$. Remark that the corresponding result for $k=1$ (parabolic problem) is sharp.


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  • 1. D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, In: ``Geometric Optics and Related Topics'' (Eds. F. Colombini & N. Lerner), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117-140. Birkhäuser, Boston, 1997.
  • 2. K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000), 85-126. MR 2001b:35031
  • 3. V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal. 34 (1998), 1005-1027. MR 99f:35079
  • 4. V. A. Galaktionov and S. I. Pohozaev, Blow-up, critical exponents and asymptotic spectra for nonlinear hyperbolic equations: Math. Preprint Univ. of Bath 00/10, 2000.
  • 5. F. John, Nonlinear wave equations, formation of singularities, University Lecture Ser. 2, Amer. Math. Soc., Providence, RI, 1990. MR 91g:35001
  • 6. V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical and angular points, Trudy Moscov. Mat. Obshch. 16 (1967), 209-292. MR 37:1777
  • 7. V. V. Kurta, On the absence of positive solutions to semilinear elliptic equations, Tr. Mat. Inst. Steklova 227 (1999), 162-169; English transl., Proc. Steklov Inst. Math. 1999, no. 4 (227), 155-162. MR 2001h:35062
  • 8. G. G. Laptev, Absence of global positive solutions for systems of semilinear elliptic inequalities in cone, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 108-124. CMP 2001:09
  • 9. G. G. Laptev, On nonexistence for a class of singular semilinear differential inequalities, Tr. Mat. Inst. Steklova 232 (2001), 223-235.
  • 10. G. G. Laptev, Nonexistence results for semilinear parabolic differential inequalities in cone, Mat. Sb., to appear.
  • 11. G. G. Laptev, Nonexistence of global solutions for higher-order evolution inequalities in unbounded cone-like domains, preprint.
  • 12. H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev. 32 (1990), 262-288. MR 91j:35135
  • 13. E. Mitidieri and S. I. Pohozaev, Absence of positive solutions for quasilinear elliptic problems on $\mathbf{R}^N$, Tr. Mat. Inst. Steklova 227 (1999), 192-222; English transl., Proc. Steklov Inst. Math. 1999, no. 4 (227), 186-216. MR 2001g:35082
  • 14. E. Mitidieri and S. I. Pohozaev, A priori estimates and nonexistence of solutions to nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001).
  • 15. S. I. Pohozaev and A. Tesei, Blow-up of nonnegative solutions to quasilinear parabolic inequalities, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), 99-109. MR 2001k:35306
  • 16. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A.P. Mikhailov, Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987; English transl., Walter de Gruyter, Berlin/New York, 1995. MR 89a:35002; MR 96b:35003
  • 17. L. Veron and S. I. Pohozaev, Blow-up results for nonlinear hyperbolic inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). 29 (2000), 393-420.
  • 18. Qi Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J. 97 (1999), 515-539. MR 2000d:35107
  • 19. Qi Zhang, Blow up and global existence of solutions to an inhomogeneous parabolic system, J. Differential Equations 147 (1998), 155-183. MR 99d:35073

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Additional Information

Gennady G. Laptev
Affiliation: Department of Function Theory, Steklov Mathematical Institute, Gubkina Street 8, Moscow, Russia
Email: laptev@home.tula.net

DOI: https://doi.org/10.1090/S1079-6762-01-00098-1
Received by editor(s): April 7, 2001
Published electronically: October 15, 2001
Additional Notes: The author was supported in part by RFBR Grant #01-01-00884.
Communicated by: Guido Weiss
Article copyright: © Copyright 2001 American Mathematical Society

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