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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
Published electronically: October 15, 2001
MathSciNet review: 1856890
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Abstract: Nonexistence of global (positive) solutions of semilinear higher-order evolution inequalities \begin{equation*} \frac {\partial ^k u}{\partial t^k}-\Delta u^m\ge |u|^q,\quad \frac {\partial ^k u}{\partial t^k}-\Delta u\ge |x|^\sigma u^q,\quad \frac {\partial ^ku}{\partial t^k}-\operatorname{div} (|x|^\alpha Du)\ge u^q \end{equation*} 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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    DelSantoGeorgievMitidieri:1997 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.
  • Keng Deng and Howard A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000), no. 1, 85–126. MR 1742850, DOI
  • Victor A. Galaktionov and Howard A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal. 34 (1998), no. 7, 1005–1027. MR 1636000, DOI
  • GalaktionovPohozaev:2000 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.
  • Fritz John, Nonlinear wave equations, formation of singularities, University Lecture Series, vol. 2, American Mathematical Society, Providence, RI, 1990. Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. MR 1066694
  • V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187
  • V. V. Kurta, On the absence of positive solutions to semilinear elliptic equations, Tr. Mat. Inst. Steklova 227 (1999), no. Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 18, 162–169 (Russian); English transl., Proc. Steklov Inst. Math. 4(227) (1999), 155–162. MR 1784314
  • Laptev:2000 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. Laptev:2001 G. G. Laptev, On nonexistence for a class of singular semilinear differential inequalities, Tr. Mat. Inst. Steklova 232 (2001), 223–235. Laptev:msb G. G. Laptev, Nonexistence results for semilinear parabolic differential inequalities in cone, Mat. Sb., to appear. Laptev:arma G. G. Laptev, Nonexistence of global solutions for higher-order evolution inequalities in unbounded cone-like domains, preprint.
  • Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055, DOI
  • È. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in ${\bf R}^N$, Tr. Mat. Inst. Steklova 227 (1999), no. Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 18, 192–222 (Russian); English transl., Proc. Steklov Inst. Math. 4(227) (1999), 186–216. MR 1784317
  • MitidieriPohozaev:book 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).
  • Stanislav I. Pohozaev and Alberto 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), no. 2, 99–109 (English, with English and Italian summaries). MR 1797514
  • SamarskiiGalaktionovKurdumovMikhailov:1987 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. ; VeronPohozaev:2000 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.
  • Qi S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J. 97 (1999), no. 3, 515–539. MR 1682987, DOI
  • Qi S. Zhang, Blow up and global existence of solutions to an inhomogeneous parabolic system, J. Differential Equations 147 (1998), no. 1, 155–183. MR 1632677, DOI

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Additional Information

Gennady G. Laptev
Affiliation: Department of Function Theory, Steklov Mathematical Institute, Gubkina Street 8, Moscow, Russia

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