Long time dynamics for damped Klein-Gordon equations

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, 1\textless{}p\textless{}(d+2)/(d-2) as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems)

The Euler Equations on Thin Domains

For the Euler equations in a thin domain Q_ε = Ω×(0, ε), Ω a rectangle in R^2, with initial data in (W^(2,q)(Qε))^3, q > 3, bounded uniformly in ε, the classical solution is shown to exist on a time interval (0, T(ε)), where T(є) → +∞ as є → 0. We compare this solution with that of a system of limiting equations on Ω

Limits of Semigroups Depending on Parameters

nuloIt is reasonable to compare dissipative semigroups with a global attractor by restricting the flows to the attractor. However, if the rate of approach to the attractor is not uniform with respect to parameters, then the transient behavior near the attractor will give more information. We introduce a concept which takes into account this transient behavior. The concept also is useful when the limit system is conservative. We give the general theory with applications to parabolic and hyperbolic PDE on thin domains as well as situations where the limit problem is conservative

Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations

In this paper, we consider the scalar reaction-diffusion equations $\partial_t u = ∆u + f(x,u,∇u)$ on a bounded domain $\Omega\subset\mathbb{R}^d$ of class $C^2$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remaining unproved