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2003 | 1 | 1 | 61-78

Tytuł artykułu

Existence and nonexistence results for reaction-diffusion equations in product of cones

Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
Problems of existence and nonexistence of global nontrivial solutions to quasilinear evolution differential inequalities in a product of cones are investigated. The proofs of the nonexistence results are based on the test-function method developed, for the case of the whole space, by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established using the method of supersolutions.

Wydawca

Czasopismo

Rocznik

Tom

1

Numer

1

Strony

61-78

Opis fizyczny

Daty

wydano
2003-03-01
online
2003-03-01

Twórcy

  • Université de La Rochelle
  • Steklov Mathematical Institute

Bibliografia

  • [1] C. Bandle, H.A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., Vol. 316 (1989), 595–622. http://dx.doi.org/10.2307/2001363
  • [2] C. Bandle, H.A. Levine, Fujita type results for convective-like reaction-diffusion equations in exterior domains, Z. Angew. Math. Phys., Vol 40 (1989), 665–676. http://dx.doi.org/10.1007/BF00945001
  • [3] C. Bandle, M. Essen, On positive solutions of Emden equations in cone-like domains, Arch. Rational Mech. Anal., Vol. 112 (1990), 319–338. http://dx.doi.org/10.1007/BF02384077
  • [4] K. Deng, H.A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., Vol. 243 (2000), 85–126. http://dx.doi.org/10.1006/jmaa.1999.6663
  • [5] H. Egnell, Positive solutions of semilinear equations in cones, Trans. Amer. Math. Soc., Vol. 330 (1992), 191–201. http://dx.doi.org/10.2307/2154160
  • [6] N. Igbida, M. Kirane, Blowup for completely coupled Fujita type reaction-diffusion system, Collocuim Mathematicum, Vol. 92 (2002), 87–96. http://dx.doi.org/10.4064/cm92-1-8
  • [7] R. Labbas, M. Moussaoui, M. Najmi, Singular behavior of the Dirichlet problem in Hlder spaces of the solutions to the Dirichlet problem in a cone, Rend. Ist. Mat. Univ. Trieste, 30, No. 1–2, (1998), 155–179.
  • [8] G.G. Laptev, The absence of global positive solutions of systems of semilinear elliptic inequalities in cones, Russian Acad. Sci. Izv. Math., Vol. 64 (2000), 108–124.
  • [9] G.G. Laptev, On the absence of solutions to a class of singular semilinear differential inequalities, Proc. Steklov Inst. Math., Vol. 232 (2001), 223–235.
  • [10] G.G. Laptev, Nonexistence of solutions to semilinear parabolic inequalities in cones, Mat. Sb., Vol. 192 (2001), 51–70.
  • [11] G.G. Laptev, Some nonexistence results for higher-order evolution inequalities in cone-like domains, Electron. Res. Announc. Amer. Math. Soc., Vol. 7 (2001), 87–93. http://dx.doi.org/10.1090/S1079-6762-01-00098-1
  • [12] G.G. Laptev, Nonexistence of solutions for parabolic inequalities in unbounded cone-like domains via the test function method, J. Evolution Equations. 2002 (in press)
  • [13] G.G. Laptev, Nonexistence results for higher-order evolution partial differential inequalities, Proc. Amer. Math. Soc. 2002 (in press).
  • [14] H.A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., Vol. 32 (1990), 262–288. http://dx.doi.org/10.1137/1032046
  • [15] H.A. Levine, P. Meier, A blowup result for the critical exponent in cones, Israel J. Math., Vol. 67 (1989), 1–7.
  • [16] H.A. Levine, P. Meier, The value of the critical exponent for reaction-diffusion equations in cones, Arch. Rational Mech. Anal., Vol. 109 (1990), 73–80. http://dx.doi.org/10.1007/BF00377980
  • [17] E. Mitidieri, S.I. Pohozaev, Nonexistence of global positive solutions to quasilinear elliptic inequalities, Dokl. Russ. Acad. Sci., Vol. 57 (1998), 250–253.
  • [18] E. Mitidieri, S.I. Pohozaev, Nonexistence of positive solutions for a system of quasilinear elliptic equations and inequalities in ℝn , Dokl. Russ. Acad. Sci., Vol. 59 (1999), 1351–1355.
  • [19] E. Mitidieri, S.I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on ℝn , Proc. Steklov Inst. Math0., Vol. 227 (1999), 192–222.
  • [20] E. Mitidieri, S.I. Pohozaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on ℝn , J. Evolution Equations, Vol. 1 (2001), 189–220. http://dx.doi.org/10.1007/PL00001368
  • [21] E. Mitidieri, S.I. Pohozaev, Nonexistence of weak solutions for some degenerate and singular hyperbolic problems on ℝn , Proc. Steklov Inst. Math., Vol. 232 (2001), 240–259.
  • [22] E. Mitidieri, S.I. Pohozaev, A priori Estimates and Nonexistence of Solutions to Nonlinear Partial Differential Equations and Inequalities, Moscow, Nauka, 2001. (Proc. Steklov Inst. Math., Vol. 234).
  • [23] S. Ohta, A. Kaneko, Critical exponent of blowup for semilinear heat equation on a product domain, J. Fac. Sci. Univ. Tokyo. Sect. IA. Math., Vol. 40 (1993), 635–650.
  • [24] S.I. Pohozaev, Essential nonlinear capacities induced by differential operators, Dokl. Russ. Acad. Sci., Vol. 357 (1997), 592–594.
  • [25] S.I. Pohozaev, 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., Vol. 11 (2000), 99–109.
  • [26] S.I. Pohozaev, A. Tesei, Critical exponents for the absence of solutions for systems of quasilinear parabolic inequalities, Differ. Uravn., Vol. 37 (2001), 521–528.
  • [27] S.I. Pohozaev, L. Veron, Blow-up results for nonlinear hyperbolic inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Vol. 29 (2000), 393–420.
  • [28] S.I. Pohozaev, L. Veron, Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group, Manuscripta math., Vol. 102 (2000), 85–99. http://dx.doi.org/10.1007/PL00005851
  • [29] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Nauka, Moscow, 1987 (in Russian). English translation: Walter de Gruyter, Berlin/New York, 1995.
  • [30] D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lect. Notes Math., Vol. 309, Springer, New York, N. Y., 1973.
  • [31] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana U. Mtah. J. 21 (1972), 979–1000. http://dx.doi.org/10.1512/iumj.1972.21.21079
  • [32] G. N. Watson, A treatise on the Theory of Bessel Functions, Cambridge University Press, London/New York, 1966.
  • [33] Qi Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., Vol. 97 (1999), 515–539. http://dx.doi.org/10.1215/S0012-7094-99-09719-3
  • [34] Qi Zhang, The quantizing effect of potentials on the critial number of reaction-diffusion equations, J. Differential Equations, Vol. 170 (2001), 188–214. http://dx.doi.org/10.1006/jdeq.2000.3815
  • [35] Qi Zhang, Global lower bound for the heat kernel of \( - \Delta + \frac{c}{{|x|^2 }}\) , Proc. Amer. Math. Soc., Vol. 129 (2001), 1105–1112. http://dx.doi.org/10.1090/S0002-9939-00-05757-9

Typ dokumentu

Bibliografia

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