Conditions for two-peaked solutions of singularly perturbed elliptic equations

We obtain necessary conditions for the existence of two-peaked solutions of singularly perturbed elliptic equations. These conditions are related to the geometry of the domain. In particular, we prove there are no two-peaked solutions in a strictly convex domain. Received: 20 January 1997 / Revised version: 2 December 1997     

Robust exponential attractors for singularly perturbed Hodgkin-Huxley equations

Here we consider a singular perturbation of the Hodgkin-Huxley system which is derived from the Lieberstein's model. We study the associated dynamical system on a suitable bounded phase space, when the perturbation parameter ε (i.e., the axon specific inductance) is sufficiently small. We prove the existence of bounded absorbing sets as well as of smooth attracting sets. We deduce the existence of a smooth global attractor Aε. Finally we prove the main result, that is, the existence of a family of exponential attractors {Eε} which is Hölder continuous with respect to ε.     

Polynomial stability of a coupled system of wave equations weakly dissipative

In this work, we consider a coupled system of wave equation. We show that the solution of this system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay. We presented a class of examples of application of the main result.     

Regularity of invariant sets in semilinear damped wave equations

Under fairly general assumptions, we prove that every compact invariant subset I of the semiflow generated by the semilinear damped wave equation

     

Nodal solutions for singularly perturbed equations with critical exponential growth

The existence and concentration behavior of nodal solutions are established for the equation −?²Δu+V(z)u=f(u) in Ω, where Ω is a domain in R², not necessarily bounded, V is a positive Hölder continuous function and f∈C¹ is an odd function having critical exponential growth.     

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.     

Strongly damped wave equations with critical nonlinearities

This note deals with the strongly damped nonlinear wave equation

_utt−Δ_ut−Δu+f(_ut)+g(u)=h$u_{tt}-\Delta u_t-\Delta u+f\left(u_t\right)+g\left(u\right)=h$

with Dirichlet boundary conditions, where both the nonlinearities f$f$ and g$g$ exhibit a critical growth, while h$h$ is a time-independent forcing term. The existence of an exponential attractor of optimal regularity is proven. As a corollary, a regular global attractor of finite fractal dimension is obtained.     

Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates

We consider the second order Cauchy problem

     

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

This paper is concerned with the existence, uniqueness and globally asymptotic stability of traveling wave fronts in the quasi-monotone reaction advection diffusion equations with nonlocal delay. Under bistable assumption, we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front (up to translation), which is monotonically increasing and globally asymptotically stable with phase shift. The influence of advection on the propagation speed is also considered. Comparing with the previous results, our results recovers and/or improves a number of existing ones. In particular, these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay, some new results are obtained.     

Scattering for the critical and localised semilinear wave equation

In this paper, we prove a scattering theorem for the critical wave equation outside convex obstacle. The proof relies on generalized Strichartz estimates.     

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.     

Local existence of solutions to dissipative nonlinear evolution equations with mixed types

In this paper, we consider the local existence of solutions to the Cauchy problems for the following nonlinear evolution equations with mixed types

     

Boundary behaviour for solutions of elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet boundary value problem for a class of second-order nonlinear elliptic partial differential equations with a gradient term and singular data. Under general conditions on the data, we study the behaviour of the solution near the boundary of the domain. Under suitable additional conditions we also investigate the second-order term in the asymptotic expansion of the solution in terms of the distance from the boundary.     

THE SINGULARLY PERTURBED NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

Abstract. The singularly perturbed problems for elliptic systems in unbounded domains are considered. Under suitable conditions and by using the comparison theorem the existence and asymptotic behavior of solution for the boundary value problems studied,     

Long-time extinction of solutions of some semilinear parabolic equations

We study the long-time behavior of solutions of semilinear parabolic equation of the following type t_∂u−Δu+a₀(x)uq=0 where , d₀>0, 1>q>0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.     

The inverse scattering problem for Schrödinger and Klein–Gordon equations with a nonlocal nonlinearity

We study the inverse scattering problem for the nonlinear Schrödinger equation and for the nonlinear Klein–Gordon equation with the generalized Hartree type nonlinearity. We reconstruct the nonlinearity from knowledge of the scattering operator, which improves the known results.     

Mildly degenerate Kirchhoff equations with weak dissipation: Global existence and time decay

We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t→+∞.We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t→+∞, with the same rate of the solution of the limit problem of parabolic type.