DCDS
Global attractor for a strongly damped wave equation with fully supercritical nonlinearities
Zhijian Yang Zhiming Liu

The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: $ u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x) $. In the case when the nonlinearities $ h(u_t) $ and $ g(u) $ are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

keywords: Strongly damped wave equation subclass of limit solutions generalized semiflow supercritical nonlinearities global attractor
DCDS
Longtime behavior of the semilinear wave equation with gentle dissipation
Zhijian Yang Zhiming Liu Na Feng
The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation: $u_{tt}-\triangle u+\gamma(-\triangle)^{\alpha} u_{t}+f(u)=g(x)$, with $\alpha\in(0,1/2)$. The main results are concerned with the relationships among the growth exponent $p$ of nonlinearity $f(u)$ and the well-posedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as $1 \leq p < p^* \equiv \frac{N + 4\alpha}{(N-2)^+}$; (ii) the subclass $\mathbb{G}$ of limit solutions has a weak global attractor as $p^* \leq p < p^{**}\equiv \frac{N+2}{N-2}\ (N \geq 3)$.
keywords: gentle dissipation exponential attractor. global attractor well-posedness Semilinear wave equation

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