Damped wave equation in the subcritical case

We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation

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Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case

Extending previous results of Oh-Zumbrun and Johnson-Zumbrun, we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type, removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions. Key to our analysis is a nonlinear cancellation estimate observed by Johnson and Zumbrun, along with a detailed understanding of the Whitham averaged system. The latter motivates a careful analysis of the Bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave, allowing us to derive the sharpened low-frequency estimates needed to close the nonlinear iteration arguments.     

Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

The complexity of wave propagation of the nonlinear Schrödinger equation with weak periodic external field

A nonlinear Schrödinger equation with external field is obtained for nonlinear optics in a non-homogeneous medium. On the basis of the Melnikov function and KAM theory, the complexity of the wave propagation of the equation is studied in the case of weak periodic external field.     

A note on parabolic equation with nonlinear dynamical boundary condition

We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate.     

Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas

In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states (λ,u)=(λ,0) at a nonlinear eigenvalue λ=λ₀ with geometric multiplicity one. Since the unilateral theorems of Rabinowitz (J. Funct. Anal. 7 (1971) 487, Theorems 1.27 and 1.40) are not true as originally stated (cf. the very recent counterexample of Dancer, Bull. London Math. Soc. 34 (2002) 533), in order to get our main results the unilateral theorem of López-Gómez (Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001, Theorem 6.4.3) is required. Our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of Rabinowitz's unilateral theory. Actually, the abstract theory developed in this paper cannot be covered with the pioneering results of Rabinowitz (1971), since in Rabinowitz's context any component of positive solutions must be unbounded, by a celebrated result attributable to Dancer (Arch. Rational Mech. Anal. 52 (1973) 181).     

Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space

In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L²-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are “good” and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave.     

Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrödinger equation, since it is well-known a solution of a linear Schrödinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable “energy function” which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.     

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.     

Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

We study the existence and nonexistence of positive (super)solutions to the nonlinear p-Laplace equation

     

Coupled nonlinear Schrödinger systems with potentials

Coupled nonlinear Schrödinger systems describe some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics and Bose-Einstein condensates. In this paper, we study the existence of concentrating solutions of a singularly perturbed coupled nonlinear Schrödinger system, in presence of potentials. We show how the location of the concentration points depends strictly on the potentials.     

Some new nonlinear inequalities and applications to boundary value problems

In this paper, we establish some new nonlinear integral inequalities of the Gronwall–Bellman–Ou-Iang-type in two variables. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations. We illustrate this by applying our new results to certain boundary value problem.     

A fourth order parabolic equation with nonlinear principal part

In this paper, we study a fourth order parabolic equation with nonlinear principal part modeling epitaxial thin film growth in two space dimensions. On the basis of the Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions.     

Degenerate nonlinear higher-order elliptic problems in domains with fine-grained boundary

We study the convergence of the solutions of the Dirichlet problem associated to a degenerate nonlinear higher-order elliptic equations in divergence form in variables domains, to a limit solution of the same type problem in a fixed domain, following the methods of the asymptotic expansion developed by Skrypnik [Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, AMS, Providence, RI, 1994] modified to weighted higher-order case.     

Existence and blow up of small-amplitude nonlinear waves with a sign-changing potential

We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential induces growth in the linearized problem, however, solutions that start out small may blow-up in finite time.     

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H² solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.     

Large time dynamics of a nonlinear spring–mass–damper model

In this paper we consider a nonlinear strongly damped wave equation as a model for a controlled spring–mass–damper system and give some results concerning its large time behaviour. It can be seen that the infinite dimensional system admits a two-dimensional attracting manifold where the equation is well represented by a classical nonlinear oscillations ODE, which can be exhibited explicitly. In contrast to other papers, this one applies Invariant Manifold Theory to a problem whose linear part is not self-adjoint.     

Reaction-diffusion systems coupled at the boundary and the Morse-Smale property

We study an one-dimensional nonlinear reaction-diffusion system coupled on the boundary. Such system comes from modeling problems of temperature distribution on two bars of same length, jointed together, with different diffusion coefficients.We prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic. To this end, we write the system as an equation with noncontinuous diffusion coefficient. We then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the Sturm-Liouville properties of the solutions of a linear elliptic problem.     

Global attractors for nonlinear wave equations with nonlinear dissipative terms

We show the existence, size and some absorbing properties of global attractors of the nonlinear wave equations with nonlinear dissipations like ρ(x,ut)=a(x)r|ut|ut.     

Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws

Nonlinear geometric optics with various frequencies for entropy solutions only in L^∞ of multidimensional scalar conservation laws is analyzed. A new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling, compactness, homogenization, and L¹-stability. New multidimensional features are recognized, especially including nonlinear propagations of oscillations with high frequencies. The validity of nonlinear geometric optics for entropy solutions in L^∞ of multidimensional scalar conservation laws is justified.