Journal of Differential Equations

Items from 1 to 15 out of 15 results

article

Neumann initial–boundary value problem for Benjamin–Ono equation

Nakao Hayashi, Elena I. Kaikina

Journal of Differential Equations > 2012 > 252 > 5 > 3520-3540

We consider the Neumann initial–boundary value problem for Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.

article

On a planar Liouville-type problem in the study of selfgravitating strings

Arkady Poliakovsky, Gabriella Tarantello

Journal of Differential Equations > 2012 > 252 > 5 > 3668-3693

Motivated by the construction of selfgravitating strings (cf. Yang, 2001, 1994 [22,23]), we analyze a Liouville-type equation on the plane, derived in Yang (1994) [23]. We establish sharp existence and uniqueness properties for the corresponding radial solutions. We investigate also when the problem allows for non-radial solutions.

article

A weighted gradient theory of phase transitions with a possibly singular and degenerate spatial inhomogeneity

Giampiero Palatucci, Enrico Valdinoci

Journal of Differential Equations > 2012 > 252 > 5 > 3381-3402

This paper studies the asymptotic behavior of a perturbed variational problem for the Cahn–Hilliard theory of phase transitions in a fluid, with spatial inhomogeneities in the internal free energy term. The inhomogeneous term can vanish or become infinite and it can also behave as an appropriate power of the distance from the boundary.The standard minimal interface criterion will be recovered even...

article

On the compatibility condition for linear systems and a factorization formula for wave functions

Alexander Sakhnovich

Journal of Differential Equations > 2012 > 252 > 5 > 3658-3667

The well-known compatibility condition for linear systems wx=Gw and wt=Fw is considered and new results are obtained. In this way, a factorization formula for wave functions, which is basic in the inverse spectral transform approach to initial–boundary value problems, is proved in greater generality than before. Some applications follow.

article

Global stability of viscous contact wave for 1-D compressible Navier–Stokes equations

Hakho Hong

Journal of Differential Equations > 2012 > 252 > 5 > 3482-3505

The viscous contact waves for one-dimensional compressible Navier–Stokes equations has recently been shown to be asymptotically stable. The stability results are called local stability or global stability depending on whether the norms of initial perturbations are small or not. Up to now, local stability results toward viscous contact waves of compressible Navier–Stokes equations have been well established...

article

Diffusive mixing of periodic wave trains in reaction–diffusion systems

Björn Sandstede, Arnd Scheel, Guido Schneider, Hannes Uecker

Journal of Differential Equations > 2012 > 252 > 5 > 3541-3574

We consider reaction–diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u0(kx−ωt;k) that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states u0(kx+ϕ±;k) as x→±∞ with different phases ϕ−≠ϕ+ at infinity for solutions that initially converge to these states as x→±∞. The proof is based on Bloch...

article

Multi-bump bound states for a system of nonlinear Schrödinger equations

Marcello Lucia, Zhongwei Tang

Journal of Differential Equations > 2012 > 252 > 5 > 3630-3657

Motivated by problems arising in nonlinear optics and Bose–Einstein condensates, we consider in RN, with N⩽3, the following system of coupled Schrödinger equations:{−Δui+λVi(x)ui=ui∑j=1dαijuj2,ui⩾0,lim|x|→∞ui(x)=0,i=1,…,d, where λ>0 is a parameter, αij=αji are positive constants, and Vi non-negative given potentials. We assume that the interior of ⋂i=1dVi−1(0) admits m connected components Ω1,…,Ωm...

article

Minimal sets in monotone and concave skew-product semiflows II: Two-dimensional systems of differential equations

Carmen Núñez, Rafael Obaya, Ana M. Sanz

Journal of Differential Equations > 2012 > 252 > 5 > 3575-3607

Recurrent nonautonomous two-dimensional systems of differential equations of ordinary, finite delay and reaction–diffusion types given by cooperative and concave vector fields define monotone and concave skew-product semiflows, whose dynamics is analyzed in this paper. A complete description of all the minimal sets in a very interesting dynamical situation is provided, and some criteria to determine...

article

On the blowup of classical solutions to the 3-D pressure-gradient systems

Ding Bingbing, Yin Huicheng

Journal of Differential Equations > 2012 > 252 > 5 > 3608-3629

The 3-D pressure-gradient system arises from the splitting of the three-dimensional compressible Euler system, which can be reduced into such a nonlinear wave equation ∂t2v−div(ev∇v)=0, where div(ev∇v)=∑i=13∂i(ev∂iv), ∂i=∂xi (1⩽i⩽3) and x=(x1,x2,x3). In this paper, we will establish a blowup result of classical solutions to the 3-D pressure-gradient systems when the initial data are of small perturbations...

article

Editorial Board

Journal of Differential Equations > 2012 > 252 > 5 > IFC

article

On ground state solutions for some non-autonomous Schrödinger–Poisson systems

Juntao Sun, Haibo Chen, Juan J. Nieto

Journal of Differential Equations > 2012 > 252 > 5 > 3365-3380

In this paper, we study the Schrödinger–Poisson system(SP){−Δu+u+K(x)ϕ(x)u=a(x)f(u),inR3,−Δϕ=K(x)u2,inR3, and prove the existence of ground state solutions for system (SP) under certain assumptions on the linear and nonlinear terms. Some recent results from different authors are extended.

article

Schaefer–Krasnoselskii fixed point theorems using a usual measure of weak noncompactness

J. Garcia-Falset, K. Latrach, E. Moreno-Gálvez, M.-A. Taoudi

Journal of Differential Equations > 2012 > 252 > 5 > 3436-3452

We present some extension of a well-known fixed point theorem due to Burton and Kirk [T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 423–431] for the sum of two nonlinear operators one of them compact and the other one a strict contraction. The novelty of our results is that the involved operators need not to be weakly continuous. Finally, an example...

article

Existence and multiplicity of positive solutions for a class of nonlinear boundary value problems

Tsung-fang Wu

Journal of Differential Equations > 2012 > 252 > 5 > 3403-3435

In this paper, we study a class of nonlinear boundary value problems in R+N. By means of minimax method and the Lusternik–Schnirelman category, the criteria of the existence, multiplicity and nonexistence of positive solutions are established.

article

On the compactness of the reduced-gravity two-and-a-half layer equations

Ran Duan, Chunhui Zhou

Journal of Differential Equations > 2012 > 252 > 5 > 3506-3519

We consider the reduced-gravity two-and-a-half model in oceanic fluid dynamics, and prove the stability of weak solutions in periodic domain Ω=T2. The proof is based on the uniform a priori estimates and the method of weak convergence, and the limit is carried out with the help of a new higher regularity estimate of the velocity, which is derived by constructing a special test function.

article

Global smooth solution for a kind of two-fluid system in plasmas

Lijia Han, Jingjun Zhang, Boling Guo

Journal of Differential Equations > 2012 > 252 > 5 > 3453-3481

A kind of two-fluid system in plasmas is studied in this paper. This system describes rich dynamics of a plasma, including the interactions between the ion acoustic wave and the plasma wave. When ignoring the effect of the high-frequency electric field, such system reduces to the ion type Euler–Poisson system. We prove that for this kind of two-fluid system, small perturbations around the equilibrium...

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Number:
5
Volume:
252
Year:
2012

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Keywords

COMPRESSIBLE NAVIER–STOKES EQUATIONS (2)
VARIATIONAL METHODS (2)
ASYMPTOTICALLY LINEAR (1)
BD ENTROPY (1)
BLOWUP (1)
COMPATIBILITY (1)
CONTACT DISCONTINUITY (1)
DECAY ESTIMATES (1)
DISPERSIVE NONLINEAR EVOLUTION EQUATION (1)
EULER–POISSON SYSTEM (1)
FACTORIZATION (1)
FUNCTIONS OF BOUNDED VARIATION (1)
FUNDAMENTAL SOLUTION (1)
GROUND STATE (1)
INITIAL–BOUNDARY VALUE PROBLEM (1)
INTEGRABLE EQUATION (1)
INVERSE SPECTRAL TRANSFORM (1)
KLAINERMAN–SOBOLEV INEQUALITY (1)
KRASNOSELSKII FIXED POINT THEOREM (1)
LACK OF COMPACTNESS (1)
LARGE INITIAL PERTURBATION (1)
LARGE TIME ASYMPTOTICS (1)
LIFESPAN (1)
LUSTERNIK–SCHNIRELMAN CATEGORY (1)
MEASURE OF WEAK NONCOMPACTNESS (1)
MULTI-BUMP SOLUTIONS (1)
NON-AUTONOMOUS SCHRÖDINGER–POISSON SYSTEM (1)
NONLINEAR BOUNDARY VALUE PROBLEMS (1)
NONLINEAR INTEGRAL EQUATIONS (1)
NONLINEAR SCHRÖDINGER SYSTEMS (1)
PHASE TRANSITIONS (1)
POSITIVE SOLUTIONS (1)
POTENTIAL WELL (1)
PRESSURE-GRADIENT SYSTEM (1)
REDUCED-GRAVITY MODEL (1)
SEMICLASSICAL LIMIT (1)
SINGULAR PERTURBATION (1)
SPATIAL INHOMOGENEITY (1)
STABILITY (1)
TWO-FLUID SYSTEM (1)
WAVE FUNCTION (1)
Γ-CONVERGENCE (1)
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