Nonexistence results for higher–order evolution partial differential inequalities

Laptev, Gennady

doi:10.1090/S0002-9939-02-06665-0

Nonexistence results for higher–order evolution partial differential inequalities
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by Gennady G. Laptev PDF

Proc. Amer. Math. Soc. 131 (2003), 415-423 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.