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Advanced Nonlinear Studies

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Volume 19, Issue 2


Blow-Up Results for Higher-Order Evolution Differential Inequalities in Exterior Domains

Mohamed Jleli / Mokhtar Kirane
  • Pôle Sciences et Technologies, Université de La Rochelle, Avenue Michel Crépeau, 17031 La Rochelle, France; and NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
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/ Bessem SametORCID iD: https://orcid.org/0000-0002-6769-3417
Published Online: 2019-02-02 | DOI: https://doi.org/10.1515/ans-2019-2040


We consider a higher-order evolution differential inequality in an exterior domain of N, N3, with Dirichlet and Neumann boundary conditions. Using a unified approach, we obtain the critical exponents in the sense of Fujita for the considered problems. Moreover, the behavior of the solutions with respect to the initial data is discussed.

Keywords: Critical Exponent; Exterior Domain; Dependence on the Initial Data; Differential Inequalities

MSC 2010: 35B33; 35B44


  • [1]

    C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc. 316 (1989), no. 2, 595–622. CrossrefGoogle Scholar

  • [2]

    R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer, Berlin, 1990, Google Scholar

  • [3]

    H. Fujita, On the blowing up of solutions of the Cauchy problem for ut=Δu+u1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124. Google Scholar

  • [4]

    V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math. 119 (1997), no. 6, 1291–1319. CrossrefGoogle Scholar

  • [5]

    R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177 (1981), no. 3, 323–340. CrossrefGoogle Scholar

  • [6]

    K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503–505. CrossrefGoogle Scholar

  • [7]

    F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), no. 1–3, 235–268. CrossrefGoogle Scholar

  • [8]

    K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977), no. 3, 407–424. CrossrefGoogle Scholar

  • [9]

    G. G. Laptev, Nonexistence of solutions to higher-order evolution differential inequalities, Sibirsk. Mat. Zh. 44 (2003), no. 1, 143–159. Google Scholar

  • [10]

    T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc. 333 (1992), no. 1, 365–378. CrossrefGoogle Scholar

  • [11]

    E. Mitidieri and S. I. Pokhozhaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklov. 234 (2001), 3–383. Google Scholar

  • [12]

    S. Pohozaev and L. Véron, Blow-up results for nonlinear hyperbolic inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 2, 393–420. Google Scholar

  • [13]

    S. Pokhozhaev, On the dependence of the critical exponent of the nonlinear heat equation on the initial function, Differ. Equ. 47 (2011), no. 7, 955–962. Web of ScienceCrossrefGoogle Scholar

  • [14]

    J. Schaeffer, The equation utt-Δu=|u|p for the critical value of p, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), no. 1–2, 31–44. Google Scholar

  • [15]

    T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations 52 (1984), no. 3, 378–406. CrossrefGoogle Scholar

  • [16]

    W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981), no. 1, 110–133. CrossrefGoogle Scholar

  • [17]

    B. T. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal. 231 (2006), no. 2, 361–374. CrossrefGoogle Scholar

  • [18]

    Q. S. Zhang, A general blow-up result on nonlinear boundary-value problems on exterior domains, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 2, 451–475. CrossrefGoogle Scholar

About the article

Received: 2018-05-07

Revised: 2018-12-28

Accepted: 2019-01-20

Published Online: 2019-02-02

Published in Print: 2019-05-01

Mohamed Jleli and Bessem Samet extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RGP-237. The research of Mokhtar Kirane is supported by NAAM research group, University of King Abdulaziz, Jeddah.

Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 2, Pages 375–390, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2019-2040.

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