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June 2003 On global solvability and asymptotic behaviour of a mixed problem for a nonlinear degenerate Kirchhoff model in moving domains
R. Benabidallah, M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira
Bull. Belg. Math. Soc. Simon Stevin 10(2): 179-196 (June 2003). DOI: 10.36045/bbms/1054818022

Abstract

In this paper we prove the existence, uniqueness and asymptotic behaviour of global regular solutions of the mixed problem for the Kirchhoff nonlinear model given by the hyperbolic-parabolic equation $$(\rho_1u_t)_t + \rho_2u_t - M\Bigl(t,\int_{\alpha(t)}^{\beta(t)} |u_x|^2dx\Bigr)u_{xx} = f \quad\text{in $\hat Q$},$$ where $\hat Q = \bigl\{(x,t)\in \Bbb R^2\bigl| \alpha(t) < x < \beta(t), 0 < t < \infty\bigr\}$ is a noncylindrical domain of $\Bbb R^2$ and $\beta(\cdot)$, $\alpha(\cdot)$ are positive functions such that $$\displaystyle\lim_{t\to\infty}(\beta(t) - \alpha(t)) = + \infty.$$ The real function $M(\cdot,\cdot)$ is such that $M(t,\lambda) \geq m_0 > 0$ $\forall (t,\lambda)\in [0,\infty[\times[0,\infty[$, while $\rho_1(\cdot)$, $\rho_2(\cdot)$ are given functions which satisfy some appropriate conditions.

Citation

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R. Benabidallah. M. M. Cavalcanti. V. N. Domingos Cavalcanti. J. Ferreira. "On global solvability and asymptotic behaviour of a mixed problem for a nonlinear degenerate Kirchhoff model in moving domains." Bull. Belg. Math. Soc. Simon Stevin 10 (2) 179 - 196, June 2003. https://doi.org/10.36045/bbms/1054818022

Information

Published: June 2003
First available in Project Euclid: 5 June 2003

zbMATH: 1038.35013
MathSciNet: MR2015197
Digital Object Identifier: 10.36045/bbms/1054818022

Subjects:
Primary: 35F30, 35K55

Rights: Copyright © 2003 The Belgian Mathematical Society

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Vol.10 • No. 2 • June 2003
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