## Nonexistence results for higher–order evolution partial differential inequalities

HTML articles powered by AMS MathViewer

- by Gennady G. Laptev
- Proc. Amer. Math. Soc.
**131**(2003), 415-423 - DOI: https://doi.org/10.1090/S0002-9939-02-06665-0
- Published electronically: September 17, 2002
- PDF | Request permission

## Abstract:

Nonexistence of global solutions to semilinear higher-order (with respect to $t$) evolution partial differential inequalities $u^{(k)}_t-\Delta u\ge |x|^\sigma |u|^q$ with $k=1,2,\dots$ in the complement of a ball is studied. The critical exponents $q^*$ are found and the nonexistence results are proved for $1<q\le q^*$. The corresponding results for $k=1$ (parabolic problem) are sharp.## References

- Serge Alinhac,
*Blowup for nonlinear hyperbolic equations*, Progress in Nonlinear Differential Equations and their Applications, vol. 17, Birkhäuser Boston, Inc., Boston, MA, 1995. MR**1339762**, DOI 10.1007/978-1-4612-2578-2 - Catherine Bandle and Howard A. Levine,
*Fujita type results for convective-like reaction diffusion equations in exterior domains*, Z. Angew. Math. Phys.**40**(1989), no. 5, 665–676 (English, with French and German summaries). MR**1013561**, DOI 10.1007/BF00945870 - C. Bandle, H. A. Levine, and Qi S. Zhang,
*Critical exponents of Fujita type for inhomogeneous parabolic equations and systems*, J. Math. Anal. Appl.**251**(2000), no. 2, 624–648. MR**1794762**, DOI 10.1006/jmaa.2000.7035 - Pierre Baras and Robert Kersner,
*Local and global solvability of a class of semilinear parabolic equations*, J. Differential Equations**68**(1987), no. 2, 238–252. MR**892026**, DOI 10.1016/0022-0396(87)90194-X - M.F. Bidaut-Veron and S.I. Pohozaev,
*Nonexistence results and estimates for some nonlinear elliptic problems*, J. Anal. Math.**84**(2001), 1–49. - Haïm Brezis and Xavier Cabré,
*Some simple nonlinear PDE’s without solutions*, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)**1**(1998), no. 2, 223–262 (English, with Italian summary). MR**1638143** - 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 10.1006/jmaa.1999.6663 - 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. - Vladimir Georgiev, Hans Lindblad, and Christopher D. Sogge,
*Weighted Strichartz estimates and global existence for semilinear wave equations*, Amer. J. Math.**119**(1997), no. 6, 1291–1319. MR**1481816** - M. Guedda and M. Kirane,
*Criticality for some evolution equations*, Differ. Uravn.**37**(2001), 610-622. - 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**, DOI 10.1090/ulect/002 - V. A. Kondrat′ev and A. A. Kon′kov,
*Properties of solutions of a class of second-order nonlinear equations*, Mat. Sb.**185**(1994), no. 9, 81–94 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math.**83**(1995), no. 1, 67–77. MR**1305756**, DOI 10.1070/SM1995v083n01ABEH003580 - V. V. Kurta,
*Certain problems of the qualitative theory of second-order nonlinear differential equations*, Doctoral (Phys.-Math.) Dissertation, Moscow: Steklov Inst. Math., Russ. Acad. Sci., 1994. - 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** - G. G. Laptev,
*Absence of global positive solutions for systems of semilinear elliptic inequalities in cone*, Izv. Ross. Akad. Nauk Ser. Mat.**64**(2000), 107–124. - G. G. Laptev,
*On the absence of solutions to a class of singular semilinear differential inequalities*, Tr. Mat. Inst. Steklova**232**(2001), 223–235. - G.G. Laptev,
*Nonexistence of solutions to semilinear parabolic inequalities in cones*, Mat. Sb.**192**(10) (2001), 51–70. - G.G. Laptev,
*Some nonexistence results for higher–order evolution inequalities in cone–like domains*, Electron. Res. Announc. Amer. Math. Soc.**7**(2001), 87–93. - Howard A. Levine,
*The role of critical exponents in blowup theorems*, SIAM Rev.**32**(1990), no. 2, 262–288. MR**1056055**, DOI 10.1137/1032046 - H. A. Levine and Q. S. Zhang,
*The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values*, Proc. Roy. Soc. Edinburgh Sect. A**130**(2000), no. 3, 591–602. MR**1769243** - E. Mitidieri and S.I. Pohozaev,
*Nonexistence of global positive solutions to quasilinear elliptic inequalities*, Dokl. Russ. Acad. Sci.**57**(1998), 250–253. - È. Mitidieri and S. I. Pokhozhaev,
*Absence of positive solutions for quasilinear elliptic problems in $\textbf {R}^N$*, 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** - E. Mitidieri and S. I. Pohozaev,
*A Priori Estimates and Blow-up of Solutions to Nonlinear Partial Differential Equations and Inequalities*, Nauka, Moscow, 2001 (Tr. Mat. Inst. Steklova**234**). - Kiyoshi Mochizuki and Ryuichi Suzuki,
*Critical exponent and critical blow-up for quasilinear parabolic equations*, Israel J. Math.**98**(1997), 141–156. MR**1459850**, DOI 10.1007/BF02937331 - Ross G. Pinsky,
*Existence and nonexistence of global solutions for $u_t=\Delta u+a(x)u^p$ in $\textbf {R}^d$*, J. Differential Equations**133**(1997), no. 1, 152–177. MR**1426761**, DOI 10.1006/jdeq.1996.3196 - S.I. Pohozaev,
*Essential nonlinear capacities induced by differential operators*, Dokl. Russ. Acad. Sci.**357**(1997), 592–594. - 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** - S. I. Pohozaev and A. Tesei,
*Instantaneous blow-up results for nonlinear parabolic and hyperbolic inequalities*, Differ. Uravn. (to appear). - S. Minakshi Sundaram,
*On non-linear partial differential equations of the hyperbolic type*, Proc. Indian Acad. Sci., Sect. A.**9**(1939), 495–503. MR**0000089** - Grozdena Todorova and Borislav Yordanov,
*Critical exponent for a nonlinear wave equation with damping*, C. R. Acad. Sci. Paris Sér. I Math.**330**(2000), no. 7, 557–562 (English, with English and French summaries). MR**1760438**, DOI 10.1016/S0764-4442(00)00228-7 - Stanislav Pohozaev and Laurent Véron,
*Blow-up results for nonlinear hyperbolic inequalities*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**29**(2000), no. 2, 393–420. MR**1784180** - Qi S. Zhang,
*Blow-up results for nonlinear parabolic equations on manifolds*, Duke Math. J.**97**(1999), no. 3, 515–539. MR**1682987**, DOI 10.1215/S0012-7094-99-09719-3 - Qi S. Zhang,
*A new critical behavior for nonlinear wave equations*, J. Comput. Anal. Appl.**2**(2000), no. 4, 277–292. MR**1793185**, DOI 10.1023/A:1010156504128 - Qi Zhang,
*A blow-up result for a nonlinear wave equation with damping: The critical case*, C. R. Acad. Sci. Paris, Série I**333**(2001), 109–114.

## Bibliographic Information

**Gennady G. Laptev**- Affiliation: Department of Function Theory, Steklov Mathematical Institute, Gubkina str. 8, Moscow, Russia
- Email: laptev@home.tula.net
- Received by editor(s): June 10, 2001
- Published electronically: September 17, 2002
- Additional Notes: The author was supported in part by INTAS project 00-0136 and RFBR Grant #01-01-00884.
- Communicated by: David S. Tartakoff
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 415-423 - MSC (2000): Primary 35G25; Secondary 35R45, 35K55, 35L70
- DOI: https://doi.org/10.1090/S0002-9939-02-06665-0
- MathSciNet review: 1933332