Dynamics for the damped wave equations on time-dependent domains

Abstract

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains Ot in R 3 are obtained from a bounded base domain O by a C 3-diffeomorphism r(•, t). Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

Pullback attractors for a weakly damped wave equation with delays and sup-cubic nonlinearity

“…Remark 2. Noticing that α = λ − C 2 > 0 in (42) is similar to the corresponding assumption in [18], which, combining with (6), will be used to obtain the existence of pullback D-absorbing sets.…”

“…In order to overcome the difficulties mentioned above, we extend the Strichartz type estimates to the non-autonomous case and establish the global well-posedness of the Shatah-Struwe solutions. Then, we verify the pullback asymptotic compactness of the process {U (t, τ )} t≥τ by constructing the energy functional and combining with the idea of contractive functions (see [17,26,28,34,36,41,42]).…”

“…Setting α = λ − C 2 > 0, we have E(ξ u (t)) ≤ E(ξ u (τ ))e −α(t−τ ) + C 1 e −α(t−τ ) + C 4 α + C 3 e −αt t τ e αs k(s) 2 2 ds, (42) for all t ≥ τ . In particular, taking t + θ instead of t, we complete the proof immediately.…”

Non-autonomous weakly damped plate model on time-dependent domains

“…This is also true in the case where the classical theory generally fails to capture the dissipation mechanism, no matter whether the external forcing function depends on time or not [7,23]. There are a great number of theoretical issues concerning the non-cylindrical domains [7,11,[15][16][17][18][20][21][22][23]26,32]. Main motivation of the study naturally comes from applications, such as the Stefan problems [25], the behavior of particles in time-dependent potential well [8], the model of tumor growth [4], and the control problems [24] etc.…”

“…Pullback attractors were investigated to the weakly damped wave equations on the moving boundary domains by using the compactness criterion [20]. Under certain assumptions on the non-cylindrical domains, the existence of pullback attractors for the weakly damped wave equations was established [32]. However, to the best of our knowledge, not much has been undertaken on the dynamics of the plate equation on such non-cylindrical domains.…”