Exponential attractor for the wave equation with structural damping and supercritical exponent (Q2828646)

The authors consider the following wave equation NEWLINE\[NEWLINEu_{tt}-\Delta u +\gamma (-\Delta)^{\alpha}u_t + f(u)=g(x)NEWLINE\]NEWLINE subjected to the initial and boundary conditions: NEWLINE\[NEWLINEu|_{\partial \Omega}=0,\quad u(x,0)=u_{0}(x),\quad u_{t}(x,0)=u_{1}(x),\quad x \in \Omega,NEWLINE\]NEWLINE in a bounded three-dimensional domain \(\Omega\) with smooth boundary \(\partial \Omega\). In this equation, \(\gamma>0\) and \(\alpha \in (1/2,1]\). The main result of the paper is the proof of existence of an exponential attractor in the natural energy space with supercritical nonlinearity. The exponential attractor is obtained using a bounded absorbing set with higher global regularity and weak quasi-stability estimates. In Section 3 of the paper, the author observes that as a consequence of Theorem 3.1, on the existence of an absorbing set with higher global regularity, the longtime dynamics of this strongly damped wave equation is of parabolic type because of the presence of the structural damping.