For general semilinear or quasilinear symmetric, hyperbolic systems, the short-wave approximation is studied by constructing approximate solutions to the initial-value problem associated with the hyperbolic operator by means of a multiscale WKB expansion. In the diffractive optics regime, the components of the leading-order terms of the approximate solutions are shown to satisfy differential equations of KP type. Whether these equations are scalar or not depends on the polarization of the initial datum and on the asymptotic behavior of the branches of the characteristic variety of the hyperbolic operator. The asymptotic stability of the approximate solution is proved. Following J.-L. Joly, G. Métivier, and J. Rauch, the cornerstone of this study is the characteristic variety of the hyperbolic operator. It is examined here in terms of perturbation theory, which yields new proofs of the so-called algebraic lemmas of geometric optics.

In this paper we consider two established structural acoustic models: Mathematically, the first model couples a hyperbolic (wave) equation, defined within a two- or three-dimensional acoustic chamber, with an elastic plate (or beam) equation, possibly with structural damping (parabolic type), defined on its elastic (flat) wall. Instead, the second model couples the same hyperbolic equation this time with a thermoelastic plate (or beam), either of parabolic type or else of {\em hyperbolic-dominated} type, defined on its flexible (flat) wall. The thermoelastic component may be supplemented by any canonical boundary conditions (B.C.), including the coupled free B.C. Moreover, its differential operators may have variable coefficients (in space). In either of the two models, coupling takes place on the elastic wall. This coupled PDE system (possibly with hyperbolic/parabolic interaction) generates a strongly continuous contraction semigroup $e^{At}$ on a natural energy space $Y$. The main result of the present paper is a backward uniqueness theorem for such structural acoustic semigroups: $e^{AT} y_0 = 0$ for some $T > 0$ and $y_0 \in Y$ implies $y_0 =0$.

We study the local and global well-posedness of the initial-value problem for the class of nonlinear dispersive PDEs of the form \[ u_t - M u_x + F(u)_x = 0, \;\;\; t\in \mathbb{R}, \] where $u=u(x,t)$, $x\in \mathbb{R} $ or $x\in \mathbb{T} $. Here $M$ is a linear operator, given in the Fourier space by the multiplication operator: $\hat{Mv} ( \xi ) = |\xi |^{2\beta } \hat{v} (\xi ),$ $ \beta \ge \frac12 $ and $F$ is a nonlinear, (sufficiently) smooth function. This equation is a generalization of the Korteweg--de Vries (KdV) equation ($\beta = 1 $), the Benjamin--Ono (BO) equation ($\beta = \frac12$) and the fifth-order KdV equation ($\beta = 2$). The nonlinearity can be very general, but a certain growth condition must be imposed in order to obtain global results. Roughly speaking, we impose that $(F'(r))_{+}$ grows at most like $\left| r\right| ^p$ as $r\to\infty $, for some $p < 4\beta $. Global existence of solutions is, therefore, intimately related to the balance between the strength of the nonlinearity and the dispersion relation. The semigroup methods developed by Goldstein--Oharu--Takahashi are being successfully applied here. Most of the results are presented in the periodic case (i.e., $x\in\mathbb{T} $), but they are also valid in the real-line case (when $x\in \mathbb{R} )$.

We consider the evolution of the density and temperature of a three-dimensional cloud of self-interacting particles. This phenomenon is modeled by a parabolic equation for the density distribution combining temperature-dependent diffusion and convection driven by the gradient of the gravitational potential. This equation is coupled with Poisson's equation for the potential generated by the density distribution. The system preserves mass by imposing a zero-flux boundary condition. Finally the temperature is fixed by energy conservation; that is, the sum of kinetic energy (temperature) and gravitational energy remains constant in time. This model is thermodynamically consistent, obeying the first and the second laws of thermodynamics. We prove local existence and uniqueness of weak solutions for the system, using a Schauder fixed-point theorem. In addition, we give sufficient conditions for global-in-time existence and blow-up for radially symmetric solutions. We do this using a comparison principle for an equation for the accumulated radial mass.

We study local and global well-posedness for the initial-value problem associated to the one-dimensional Schrödinger-Boussinesq equations in low regularity spaces. To establish these results we make use of sharp $L^p$-$L^q$ estimates.

By explicitly bounding the growth of terms in a singular perturbation expansion with a small parameter ${\varepsilon}$, we show that it is possible to find a solution that satisfies a balance relation (which defines the slow manifold) up to an error that scales exponentially in ${\varepsilon}$ as ${\varepsilon}\to0$. This is first done for a generic finite-dimensional dynamical system with polynomial nonlinearity, followed by a continuous fluid case. In addition, for the finite-dimensional system, we show that, properly initialized, the solution of the full model stays within an exponential distance to that of the balance equation (i.e., evolution on the slow manifold) over a timescale of order one (independent of ${\varepsilon}$).

We study the existence of multiple positive solutions of $ -\Delta u = \lambda u^{-q} +u^p $ in $\Omega$ with homogeneous Dirichlet boundary condition, where $\Omega$ is a bounded domain in $\mathbb R^N$, $\lambda >0$, and $0 < q \leq 1 < p \leq (N+2)/(N-2)$. We show by a variational method that if $\lambda$ is less than some positive constant then the problem has at least two positive, weak solutions including the cases of $q=1$ or $p=(N+2)/(N-2)$. We also study the regularity of positive weak solutions.

By exploiting a variational identity of Pohožaev-Pucci-Serrin type for solutions of class $C^1$, we get some necessary conditions for locating the peak-points of a class of singularly perturbed quasilinear elliptic problems in divergence form. More precisely, we show that the points where the concentration occurs, in general, must belong to what we call the set of weak-concentration points. Finally, in the semilinear case, we provide a new necessary condition which involves the Clarke subdifferential of the ground-state function.