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Global Nonexistence for Nonlinear Kirchhoff Systems

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Abstract

In this paper we consider the problem of non-continuation of solutions of dissipative nonlinear Kirchhoff systems, involving the p(x)-Laplacian operator and governed by nonlinear driving forces f = f (t, x, u), as well as nonlinear external damping terms Q = Q(t, x, u, u t ), both of which could significantly dependent on the time t. The theorems are obtained through the study of the natural energy Eu associated to the solutions u of the systems. Thanks to a new approach of the classical potential well and concavity methods, we show the nonexistence of global solutions, when the initial energy is controlled above by a critical value; that is, when the initial data belong to a specific region in the phase plane. Several consequences, interesting in applications, are given in particular subcases. The results are original also for the scalar standard wave equation when p ≡ 2 and even for problems linearly damped.

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Authors and Affiliations

  1. Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze, Viale G.B. Morgagni 67/A, 50134, Firenze, Italy

    Giuseppina Autuori

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, Via Vanvitelli 1, 06123, Perugia, Italy

    Patrizia Pucci & Maria Cesarina Salvatori

Authors
  1. Giuseppina Autuori
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  2. Patrizia Pucci
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  3. Maria Cesarina Salvatori
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Correspondence to Giuseppina Autuori.

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Communicated by A. Bressan

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Autuori, G., Pucci, P. & Salvatori, M.C. Global Nonexistence for Nonlinear Kirchhoff Systems. Arch Rational Mech Anal 196, 489–516 (2010). https://doi.org/10.1007/s00205-009-0241-x

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