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Register a new userAttractors for damped quintic wave equations in bounded domains
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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.
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We report on new results concerning the global well-posedness, dissipativity and attractors of the damped quintic wave equations in bounded domains of R^3.
Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors of finite Hausdorff and fractal dimension is verified.
The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in $mathbb{R}^3$ with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.
We give a detailed study of attractors for measure driven quintic damped wave equations with periodic boundary conditions. This includes uniform energy-to-Strichartz estimates, the existence of uniform attractors in a weak or strong topology in the energy phase space, the possibility to present them as a union of all complete trajectories, further regularity, etc.
Consider the integral equation begin{equation*} f^{q-1}(x)=int_Omegafrac{f(y)}{|x-y|^{n-alpha}}dy, f(x)>0,quad xin overline Omega, end{equation*} where $Omegasubset mathbb{R}^n$ is a smooth bounded domain. For $1<alpha<n$, the existence of energy maximizing positive solution in subcritical case $2<q<frac{2n}{n+alpha}$, and nonexistence of energy maximizing positive solution in critical case $q=frac{2n}{n+alpha}$ are proved in cite{DZ2017}. For $alpha>n$, the existence of energy minimizing positive solution in subcritical case $0<q<frac{2n}{n+alpha}$, and nonexistence of energy minimizing positive solution in critical case $q=frac{2n}{n+alpha}$ are also proved in cite{DGZ2017}. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as $qto (frac{2n}{n+alpha})^+ $ (in the case of $1<alpha<n$), and the blowup behaviour of energy minimizing positive solution as $qto (frac{2n}{n+alpha})^-$ (in the case of $alpha>n$) are analyzed. We see that for $1<alpha<n$ the blowup behaviour obtained is quite similar to that of the elliptic equation involving subcritical Sobolev exponent. But for $alpha>n$, different phenomena appears.
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