Attractors for Damped Quintic Wave Equations in Bounded Domains

Abstract

Abstract. The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.

Optimal control of an energy-critical semilinear wave equation in 3D with spatially integrated control constraints

“…Another consequence of the energy conservation for Shatah-Struwe and, per Proposition 2.4, mild solutions, is uniqueness: [24,Cor. 3.4]).…”

Uniform attractors for measure-driven quintic wave equations

“…As we have already mentioned, the global existence result for Shatah-Struwe solutions based on the non-concentration arguments (and stated in Proposition 4.6) does not give any control of the Strichartz norm u L 4 (T,T +1;L 12 ) in terms of T and the corresponding norms of the initial data and the external forces. In particular, we do not have any control of the behaviour of this norm as T → ∞ which in turn leads to essential problems in the attractor theory, see [20] for the details. The aim of this Section is to estimate this Strichartz norm in terms of the energy norm and the proper norm of the external forces.…”

“…Energy-to-Strichartz estimate (1.3) allows us to deduce the control and establish the dissipativity of u in the Strichartz norm based on the standard energy estimate. Since the control of this norm is enough for the uniqueness, the obtained control gives the well-posedness, dissipativity and the existence of global/uniform attractors in the way which is similar to the clasical cubic case, see [14], [22] and [20] for the case of R 3 , T 3 and a bounded domain endowed with the Dirichlet boundary conditions respectively (see also [32] for the case of damped wave equations with fractional damping). In contrast to this, very few is known about the solutions of (1.1) in the supercritical (superquintic) growth rate of the non-linearity f .…”

A note on the global attractor for weakly damped wave equation

“…[1][2][3][4]), and the global attractor has been established in the natural phase space under the conditions (1.2) and (1.3). Most recently, in [5], the long-time behavior of the Shatah-Struwe solution of damped quintic wave equation has been considered, the existence of a global attractor has been proved, due to the recent progress in Strichartz estimates for the case of bounded domains (see, e.g. [6]).…”

“…However, for the weakly damped wave equation (1.1), it seems difficult to apply the corresponding method to verify asymptotic compactness of the solution semigroup, which is a key point to obtain the existence of attractor. Motivated by [5], in this note, we present a new method about decomposition of the solution to Eq. (1.1) together with the Strichartz estimate to the wave equation (see [6]) to verify asymptotic compactness of the solution semigroup.…”