On global solvability and asymptotic behaviour of a mixed problem for a nonlinear degenerate Kirchhoff model in moving domains

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.