Our aim in this article is to construct exponential attractors for nonautonomous partially dissipative systems by adapting to the nonautonomous case the method presented in [1] and based on a decomposition of the difference of two trajectories. We follow the ideas developed in [2] which consist in studying an equation on an extended space. As an example, we prove the existence of exponential attractors for the slightly compressible Navier-Stokes equations with quasiperiodic in time volume forces.

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the higher-order dispersive equations of Korteweg-de Vries type with weak nonlinearity (wKdV): $$ u_t + \partial_x f(u) + \mathcal{K} u = 0, $$ where $x, t \in \mathbf {R}$, $f(u) = |u|^{\rho -1}u$ if $\rho > \nu$ or $f(u) = u^{\rho }$ if $\rho > \nu$ is integer, the operator $\mathcal{K} $ is a pseudodifferential operator with a homogeneous and conservative symbol $K(p)$ of order $\nu > 3$, namely, $\mathcal{K}u = \mathcal{F}^{-1} K(p) \hat u(p), K(p) = - \frac{i}{\nu} |p|^{\nu - 1} p, $ $\mathcal{F}\phi$ or $\hat \phi$ is the Fourier transformation of $\phi$ and $\mathcal{F}^{-1} \phi $ is the inverse Fourier transformation of $\phi$. If the power $\rho$ of the nonlinearity is greater than $\nu$, then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. More precisely, we show that the solution $u(t)$ satisfies the decay estimate $$ \|u(t)\|_{L^\beta} \le C(1 + t)^{-\frac{1}{\nu}(1-\frac{1}\beta)} \quad \text{for} \quad \beta \in (\frac{2\nu -2}{\nu - 2},\infty], $$ $$ \|uu_x(t)\|_{L^\infty} \le Ct^{-2/\nu}(1+t)^{-1/\nu} $$ and using these estimates we prove the existence of the scattering state $u_+\in L^2$ such that $$ \|u(t) - U(t)u_+\|_{L^2} \le Ct^{-\frac{\rho -\nu}{\nu}} \quad \text{and} \quad \|u(t) - U(t)u_+\|_{L^\infty} \le Ct^{-\frac{1+\rho -\nu}{\nu}} $$ for any small initial data belonging to the weighted Sobolev space $H^{1,1} = \{ \phi \in L^2; \|(1+|x|^2)^{1/2}(1-\partial_x^2)^{1/2} \phi\|_{L^2}<\infty\}$, where $U(t)$ is the free evolution group, associated with corresponding linear equation.

For a positive, strongly continuous evolution family $\mathcal{U}$ on $L^p(\Omega)$ and a positive, measurable, time dependent potential $V(\cdot)$ we construct a corresponding absorption evolution family $\mathcal {U}_V$ by a procedure introduced by J. Voigt, [24, 25], in the autonomous case. We give sufficient conditions on $\mathcal{U}$ and $V$ such that $\mathcal {U}_V$ is strongly continuous and satisfies variation of constants formulas. For an evolution family $\mathcal {U}$ on $L^1(\mathbb{R}^N)$ satisfying upper and lower Gaussian estimates exponential stability of $\mathcal {U}_V$ is characterized by a condition on the size of $V$ extending recent results by W.Arendt and C.J.K. Batty, [2, 3, 6], in the autonomous case and by D. Daners, M. Hieber, P. Koch Medina, and S. Merino, [8, 12], in the time periodic case. An application to a second order parabolic equation with real coefficients and singular potential is given.

We prove a strong continuity result for the relative rearrangement map. This new result and its corollary are used for the resolution of equations of the form $-\Delta u = F(u)$ via a Brouwer fixed point theorem. The nonlocal nonlinearity $F$ might depend on the monotone rearrangement $u_*$, its derivative $u'_*$ and the relative rearrangement of $u$ with respect to the data.

A partial regularity theorem is established for weak solutions of elliptic equations of the form $\mbox{div}(A(y)\nabla\psi)=0$. Here we allow the possibility that the eigenvalues of $A(y)$ are not bounded away from $0$ below. This result is then used to prove an everywhere regularity theorem for weak solutions of the initial- boundary-value problem for the system $\frac{\partial u}{\partial t}-\Delta u = \sigma(u)|\nabla \varphi|^2$, $\mbox{div}(\sigma(u) \nabla\varphi)=0$ in the case where $\sigma$ may decay exponentially.

We consider a mathematical model of the noise reduction problem, which couples two hyperbolic equations: the wave equation in the interior ("chamber")---which describes the unwanted acoustic waves---and a (hyperbolic) Kirchoff equation ---which models the vibrations of the elastic wall. In past models, the elastic wall was modeled by an Euler-Bernoulli equation with Kelvin-Voight damping (parabolic model). Our main result is a sharp regularity result, in two dual versions, of the resulting system of two coupled hyperbolic P.D.E.'s. With this regularity result established, one can then invoke a wealth of abstract results from [14], [15], [16], [19], etc. on optimal control problems, min-max game theory (and $H^\infty$-problems), etc. The proof of the main result is based on combining technical results from [18] and [11].