Dynamics of wave equations with moving boundary

Abstract

This paper is concerned with long-time dynamics of weakly damped semilinear wave equations defined on domains with moving boundary. Since the boundary is a function of the time variable the problem is intrinsically non-autonomous. Under the hypothesis that the lateral boundary is time-like, the solution operator of the problem generates an evolution process U (t, τ) : X τ → X t , where X t are timedependent Sobolev spaces. Then, by assuming the domains are expanding, we establish the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the forcing terms. Our assumptions allow nonlinear perturbations with critical growth and unbounded time-dependent external forces.

On the 1d wave equation in time-dependent domains and the problem of debond initiation

“…For an account on classical results in arbitrary dimension, without claim to be exhaustive we refer to e.g. [6,7,19,21,27,30,34]; see also [28,37] and references therein for recent applications. An abstract approach allowing for irregular time-dependence was proposed in [1], still assuming that the domains are not degenerate and that initial conditions are given.…”

Dynamics for the damped wave equations on time-dependent domains

“…For example, under the assumption that there exists a one-to-one mapping φ : Q → Q * of class C 3 , with bounded derivatives and inverse ψ of class C 1 and satisfying (3.1) in [20], the existence and uniqueness of global weak solutions was proved by Cooper and Bardos [20]. Very recently, under the hypothesis that the lateral boundary is time-like and the domains are expanding, Ma et al [37] established the pullback attractors of weakly damped wave equations by presented a useful compactness criterion.…”

“…In the present paper, we establish the existence of a pullback attractor to the problem (4) under the assumption that the time-varying domains obtained by a temporally continuous dependent diffeomorphic transformation of a bounded reference domain. Our work is in the spirit of [20,22,31,32,33,37], and many ideas of this article are taken from these works. However, we develop several generalisations, which can be regarded as the main features of this paper.…”

“…We also construct a sufficient condition to ensure the transformation is hyperbolic in Appendix which may be of independent interest. Secondly, compared with [37], our domains are more complicated and general, which brought some intrinsic difficulties in our process, such as obtain the strong solutions. Hence, much of our effort is in proving the existence and uniqueness of strong and weak solutions in appropriate functions spaces as well as in establishing energy inequalities.…”

“…Thirdly, the nonlinearity g of equation 4has a critical growth rate, i.e., the mapping g from H 1 0 to L 2 is continuous, but not compact (see [43] and the references therein). To overcome the difficulties brought by the critical nonlinearity, usually, the socalled energy method developed by Ball [7], a decomposition technique (e.g., see [3,46]) and the "contractive function" method (see [18], [29] and [37] for further details and some extensions) have been successfully applied to prove the asymptotic compactness of the solutions. Due to the domain is time-dependent, we combine the two ideas presented in [31] (see Section 4.1) and [29] to prove the process is D-pullback asymptotic compact.…”