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 Title Longtime dynamics for the strongly damped wave equation with critical and supercritical nonlinearities
 Name Zhijian Yang Country Peoples Rep of China Email yzjzzvt@zzu.edu.cn Co-Author(s) Submit Time 2014-02-06 22:17:12 Session Special Session 37: Global or/and blowup solutions for nonlinear evolution equations and their applications
 Contents We are concerned with the longtime dynamics of the Kirchhoff equation with strong dissipation $u_{tt}-M(\|\nabla u\|^2)\Delta u- \Delta u_t+h(u_t)+g(u)=f(x)$. We prove that the IBVP of the equation admits a unique global weak solution, the related dynamical system has a finite dimensional global attractor and an exponential attractor provided that $h(s)$ is quasi-monotone and both growth exponents $p$ and $q$ of the nonlinearities $h(s)$ and $g(s)$ are up to critical range, that is, $1\leq p, q\leq p^*=\frac{N+2}{(N-2)^+}$. Moreover, the optimal regularity of the global attractor is established within further restrictions that the above-mentioned growth exponents are equal and up to subcritical range. In particular, when $M(s)=1$, we further show that the IBVP of the equation possesses a global weak solution provided that the growth exponents $p$ and $q$ are fully supercritical, that is, $p=q>p^*$, and the related generalized semiflow has in natural energy space endowed with strong topology a global attractor.