American Institute of Mathematical Sciences

u_tt$ + a u_t + A u = f(u), \ t>0, $
$u(0)=u_0\in H^1_0(\Omega), \ \ u_t(0)=v_0\in L^2(\Omega),$

where $f:\R\to\R$ is a continuously differentiable function satisfying the growth condition $|f(s)-f(t)|\leq C|s-t|(1+|s|^{\rho-1}+|t|^{\rho-1})$, $1<\rho<\frac{N+2}{N-2}$, ($N\geq 3$), and the dissipativeness condition $\lim$sup$_|s|\to\infty \frac{f(s)}{s}< \lambda_1$ with $\lambda_1$ being the first eigenvalue of $A$. We construct the global weak solutions of this problem as the limits as $\eta\to0^+$ of the solutions of wave equations involving the strong damping term $2\eta A^{1/2} u$ with $\eta>0$. We define a subclass $\mathcal LS\subset C([0,\infty),L^2(\Omega)\times H^{-1}(\Omega))\cap L^\infty([0,\infty),H^1_0(\Omega)\times L^2(\Omega))$ of the 'limit' solutions such that through each initial condition from $H^1_0(\Omega)\times L^2(\Omega)$ passes at least one solution of the class $\mathcal LS$. We show that the class $\mathcal LS$ has bounded dissipativeness property in $H^1_0(\Omega)\times L^2(\Omega)$ and we construct a closed bounded invariant subset A of $H^1_0(\Omega)\times L^2(\Omega)$, which is weakly compact in $H^1_0(\Omega)\times L^2(\Omega)$ and compact in $H^s_{\I}(\Omega)\times H^{s-1}(\Omega)$, $s\in[0,1)$. Furthermore A attracts bounded subsets of $H^1_0(\Omega)\times L^2(\Omega)$ in $H^s_\{I\}(\Omega)\times H^{s-1}(\Omega)$, for each $s\in[0,1)$. For $N=3,4,5$ we also prove a local uniqueness result for the case of smooth initial data.

$ u(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} v^q (y) dy$
$ v(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} u^p(y) dy$

with $0 < \mu < n$. Under some integrability conditions, we obtain radial symmetry of positive solutions by using the method of moving planes in integral forms. In the special case when $\mu = 2$, we show that the integral system is equivalent to the elliptic PDE system

-Δ $u = v^q (x)$
-Δ $v = u^p (x)$

in $R^n$. Our symmetry result, together with non-existence of radial solutions by Mitidieri [30], implies that, under our integrability conditions, the PDE system possesses no positive solution in the subcritical case. This partially solved the well-known Lane-Emden conjecture.

∂_tt $u - \Delta u = -u^{3} $
$u(0,x) = u_{0}(x) $
$\partial_{t} u(0,x) = u_{1}(x)$

with data $( u_{0}, u_{1} ) \in H^{s} \times H^{s-1}$, $1 > s > \frac{13}{18} $≈ 0.722. The main task is to estimate the variation of an almost conserved quantity on an arbitrary long time interval. We divide it into subintervals. On each of these subintervals we write the solution as the sum of its linear part adapted to the subinterval and its corresponding nonlinear part. Some terms resulting from this decomposition have a controlled global variation and other terms have a slow local variation.