On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential.     

Blow up for nonlinear parabolic equations with time degeneracy under dynamical boundary conditions

We study the blow up behaviour of nonlinear parabolic equations including a time degeneracy, under dynamical boundary conditions. For some exponential and polynomial degeneracies, we develop some energy methods and some spectral comparison techniques and derive upper bounds for the blow up times.     

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton–Jacobi–Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles–Souganidis (2000) for first-order Hamilton–Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.     

Transversality in scalar reaction-diffusion equations on a circle

We prove that stable and unstable manifolds of hyperbolic periodic orbits for general scalar reaction-diffusion equations on a circle always intersect transversally. The argument also shows that for a periodic orbit there are no homoclinic connections. The main tool used in the proofs is Matano's zero number theory dealing with the Sturm nodal properties of the solutions.     

Well-posedness results for triply nonlinear degenerate parabolic equations

We study well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problems of the kind

     

Approaching a vertex in a shrinking domain under a nonlinear flow

We consider here the homogeneous Dirichlet problem for the equation , in a noncylindrical domain in space-time given by . By means of matched asymptotic expansion techniques we describe the asymptotics of the maximal solution approaching the vertex x=0, t=T, in the three different cases p>1/2, p=1/2(vertex regular), p<1/2 (vertex irregular).     

On the convergence of global and bounded solutions of some evolution equations

It is shown that a special case of the well-known Lojasiewicz gradient inequality is sufficient to give a unified background for many convergence results in gradient or gradient-like systems appearing previously in the Literature. Besides as an illustration we give a direct proof of convergence in the case of 1D wave equations by a suitable adaptation of Zelenyak’s method.     

Obstacle problem for nonlinear parabolic equations

We show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time-dependent obstacle. The solution is constructed by an adaptation of the Schwarz alternating method. Moreover, if the obstacle is Hölder continuous, we prove that the solution inherits the same property.     

Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems

Global asymptotic dynamics of a representative cubic-autocatalytic reaction-diffusion system, the reversible Selkov equations, are investigated. This system features two pairs of oppositely signed nonlinear terms so that the asymptotic dissipative condition is not satisfied, which causes substantial difficulties in an attempt to attest that the longtime dynamics are asymptotically dissipative. An L² to H¹ global attractor of finite fractal dimension is shown to exist for the semiflow of the weak solutions of the reversible Selkov equations with the Dirichlet boundary condition on a bounded domain of dimension n≤3. A new method of rescaling and grouping estimation is used to prove the absorbing property and the asymptotical compactness. Importantly, the upper semicontinuity (robustness) in the H¹ product space of the global attractors for the family of solution semiflows with respect to the reverse reaction rate as it tends to zero is proved through a new approach of transformative decomposition to overcome the barrier of the perturbed singularity between the reversible and non-reversible systems by showing the uniform dissipativity and the uniformly bounded evolution of the union of global attractors under the bundle of reversible and non-reversible semiflows.     

Stability of perturbed nonlinear parabolic equations with Sturmian property

We study stability of an equilibrium f∗ of autonomous dynamical systems under asymptotically small perturbations of the equation. We show that such stability takes place if the domain of attraction of the equilibrium f∗ contains a one-parametric ordered family . In the stability analysis we need a special S-relation (a kind of “restricted partial ordering”) to be preserved relative to the family . This S-relation is inherited from the Sturmian zero set properties for linear parabolic equations. As main applications, we prove stability of the self-similar blow-up behaviour for the porous medium equation, the p-Laplacian equation and the dual porous medium equation in with nonlinear lower-order perturbations. For such one-dimensional parabolic equations the S-relation is Sturm's Theorem on the nonincrease of the number of intersections between the solutions and particular solutions with initial data in . This Sturmian property plays a key role and is true for the unperturbed PME, but is not true for perturbed equations.     

Critical curves of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

In this paper, we consider the existence and non-existence of global solutions of the non-Newtonian polytropic filtration equations with nonlinear boundary conditions. We first obtain the critical global existence curve by constructing various self-similar supersolutions and subsolutions. And then the critical Fujita curve is conjectured with the aid of some new results.     

Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation

ut−diva(x,∇u)+f(x,u)=0     

Time-periodic solutions of the time-dependent Ginzburg-Landau equations of superconductivity

The problem of existence of time-periodic solutions for the time-dependent Ginzburg-Landau equations of superconductivity is discussed. It is assumed that the applied magnetic field H is -periodic in time, and so is the associated dynamical process. We prove the existence of -periodic solutions in time, which are exactly the fixed points of the associated period mapping.     

Existence of solutions of the very fast diffusion equation in bounded and unbounded domain

We prove the existence of a unique solution of the following Neumann problem , u > 0, in (a, b) × (0, T), u(x, 0) = u ₀(x) ≥ 0 in (a, b), and , where if m < 0, if m = 0, and m≤ 0, , and the case −1 < m ≤ 0, , for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity.     

Viscosity method for homogenization of parabolic nonlinear equations in perforated domains

In this paper, we develop a viscosity method for homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of the capacity of columns or the capacity of punctured ball is too high or too small, the limit of u? will converge to trivial solutions. The critical decay rates of having nontrivial solution are obtained with the construction of barriers. We also show the limit of u? satisfies a homogenized equation with a term showing the effect of the highly oscillating obstacles or perforated domain in viscosity sense.     

Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing

Summary. We propose and prove a convergence of the semi-implicit finite volume approximation scheme for the numerical solution of the modified (in the sense of Catté, Lions, Morel and Coll) Perona–Malik nonlinear image selective smoothing equation (called anisotropic diffusion in the image processing). The proof is based on a-priori estimates and Kolmogorov's compactness theorem. The implementation aspects and computational results are discussed. Received January 7, 1999 / Revised version received May 31, 2000 / Published online March 20, 2001