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On a class of semilinear weakly hyperbolic equations

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Abstract

In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations

$$ u_{tt}-a_{\lambda_1}(t)\Delta_x u=-a_{\lambda_2}(t)|u|^{p-1}u. $$

Here u=u(x,t), \(x\in \mathbb{R}^n\) and for λ≥ 0, a λ ≥ 0 is a continuous function that behaves as |t–t0|λ close to some t0>0. We conjecture the existence of a critical exponent p c (λ1,λ2,n) such that for p≤ p c (λ1,λ2,n) a global existence theorem holds. For suitable λ1,λ2,n, we recall some known results and add new ones.

Keywords: Critical exponents for semilinear equations, Weak hyperbolicity

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References

  • 1. Brenner, P., Kumlin, P.: On wave equations with supercritical nonlinearities. Arch. Math. 74, 129–147 (2000)

    Google Scholar 

  • 2. Capuzzo Dolcetta, I., Cutrì, A.: On the Liouville Property for the Sublaplacians. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25, 239-256 (1997)

    Google Scholar 

  • 3. D'Ambrosio, L., Lucente, S.: Nonlinear Liouville Theorems for Grushin and Tricomi Operators. J. Differential Equations 123, 511-541 (2003)

    Google Scholar 

  • 4. D'Ancona, P.: A Note on a Theorem of Jörgens. Math. Z. 218, 239-252 (1995)

    Google Scholar 

  • 5. D'Ancona, P., Di Giuseppe, A.: Global Existence with Large Data for a Nonlinear Weakly Hyperbolic Equation. Math. Nachr. 231, 5-23 (2001)

  • 6. Fanelli, L., Lucente, S.: The critical case for a semilinear weakly hyperbolic equation. El. Journ. of Diff. Eq. 101, 1-13 (2004)

    Google Scholar 

  • 7. Jörgens, K.: Das Anfangswertproblem in Großem für eine Klasse nichtlinearen Wellengleichungen. Math. Z. 77, 295-308 (1961)

  • 8. Lebeau, G.: Perte de régularité pour les équations d'ondes sur-critiques. Bull. Soc. Math. France 133, 145–157 (2005)

    Google Scholar 

  • 9. Rauch, J.: The u5-Klein-Gordon Equation. In Nonlinear partial differential equations and their applications. Coll. de France Semin., Vol. I, Res. Notes Math. Pitman Research Notes in Math. 53, 335-364 (Pitman 1981)

  • 10. Reissig, M.: L p -L q decay estimates for wave equations with time-dependent coefficients. J. Nonlinear Math. Phys. 11, 534–548 (2004)

  • 11. Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemitskij operators and Nonlinear Partial Differential Equations. Walther de Gryter, Berlin (1996)

  • 12. Shatah, J., Struwe, M.: Geometric Wave Equations. Courant Lecture Notes in Mathematics, New York (1998)

  • 13. Wirth, J.: Asymptotic properties of the solutions to wave equations with time-dependent dissipation. PhD thesis, TU Bergakademie Freiberg (2004)

  • 14. Yagdjian, K.: A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain. J. Differential Equations 206, 227–252 (2004)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bari, via E. Orabona 4, I-70125 Bari, Italy, Tel.: 0805442275, Fax: 0805963612,  

    Sandra Lucente

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  1. Sandra Lucente
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Lucente, S. On a class of semilinear weakly hyperbolic equations. Ann. Univ. Ferrara 52, 317–335 (2006). https://doi.org/10.1007/s11565-006-0024-3

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  • DOI: https://doi.org/10.1007/s11565-006-0024-3

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