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Global solution and asymptotic behaviour of nonlinear Kirchhoff model in infinitely increasing moving domains

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In questo lavoro si studia l'esistenza, l'unicità e il comportamento asintotico della soluzione globale per l'equazione iperbolico-parabolica non lineare del tipo Kirchhoff-Carrieru tt + μu t-M(∫Ωt|∇u|2 dx)δu=0 in\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Q} \) dove\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Q} \) :={(x,t∈ℝ3|x=yγ(t),(t,y)∈]0, ∞[xΩ}è un dominio non cilindrico di ℝ3, μ è una costante positiva eγ(t) è una funzione positiva tale che\(\mathop {\lim }\limits_{t \to \infty } \gamma (t) = + \infty \). La funzioneM(·) soddisfaM(r)≥m 0 >0 ∀r∈[0, ∞[.

Abstract

In this paper we study the existence, uniqueness and asymptotic behaviour of global solutions for the nonlinear hyperbolic-parabolic equation of Kirchhoff-Carrier typeu tt + μu t-M(∫Ωt|∇u|2 dx)δu=0 in\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Q} \), where\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Q} \) :={(x,t∈ℝ3|x=yγ(t),(t,y)∈]0, ∞[xΩ} is a noncylindrical domain of ℝ3, μ is a positive constant andγ(t) is a positive function such that\(\mathop {\lim }\limits_{t \to \infty } \gamma (t) = + \infty \). The real functionM(·) is such thatM(r)≥m 0 >0 ∀r∈[0, ∞[.

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Author notes

  1. Rachid Benabidallah

    Present address: Dipartimento di Matematica, Università di Torino, Via C. Alberto 10, 10123, Torino, Italy

  2. François Ebobisse

    Present address: Dipartimento di Matematica, “L. Tonelli”, Università di Pisa, Via F. Buonarrotti 2, 56127, Pisa, Italy

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    1. Rachid Benabidallah
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    2. François Ebobisse
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    Benabidallah, R., Ebobisse, F. Global solution and asymptotic behaviour of nonlinear Kirchhoff model in infinitely increasing moving domains. Ann. Univ. Ferrara 47, 207–229 (2001). https://doi.org/10.1007/BF02838183

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