
ISSN:
1078-0947
eISSN:
1553-5231
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Discrete & Continuous Dynamical Systems - A
January 2020 , Volume 40 , Issue 1
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We consider the resonant system of amplitude equations for the conformally invariant cubic wave equation on the three-sphere. Using the local bifurcation theory, we characterize all stationary states that bifurcate from the first two eigenmodes. Thanks to the variational formulation of the resonant system and energy conservation, we also determine variational characterization and stability of the bifurcating states. For the lowest eigenmode, we obtain two orbitally stable families of the bifurcating stationary states: one is a constrained maximizer of energy and the other one is a local constrained minimizer of the energy, where the constraints are due to other conserved quantities of the resonant system. For the second eigenmode, we obtain two local constrained minimizers of the energy, which are also orbitally stable in the time evolution. All other bifurcating states are saddle points of energy under these constraints and their stability in the time evolution is unknown.
Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [
We prove the unconditional uniqueness of solutions to the derivative nonlinear Schrödinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS into a new equation (the so-called normal form equation) for which nonlinear estimates can be easily established in
Let
We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem
with an integro-differential operator, including the fractional Laplacian, of the form
when
We deal with a multiparameter Dirichlet system having the form
where
In the present paper, we show that an analogous N-barrier maximum principle (see [
We continue and supplement studies from [G. Karch and X. Zheng, Discrete Contin. Dyn. Syst. 35 (2015), 3039-3057] on solutions to the three dimensional incompressible Navier-Stokes system which are regular outside a curve in
We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map
This paper deals with the derivation of entropy solutions to Cauchy problems for a class of scalar conservation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes which are product of a function of the density
We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus
Let
In this paper, we explore the metastable states of nonlinear Klein-Gordon equations with potentials. These states come from instability of a bound state under a nonlinear Fermi's golden rule. In [
This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carré du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in
We investigate a distinguished low Mach and Rossby - high Reynolds and Péclet number singular limit in the complete Navier-Stokes-Fourier system towards a strong solution of a geostrophic system of equations. The limit is effectuated in the context of weak solutions with ill prepared initial data. The main tool in the proof is based on the relative energy method.
In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schrödinger equations
We present an example of a
This paper is concerned with the existence/nonexistence of positive solutions of a weighted Hardy-Littlewood-Sobolev type integral system. Such a system is related to the extremal functions of the weighted Hardy-Littlewood-Sobolev inequality. The Serrin-type condition is critical for existence of positive solutions in
We investigate the relations between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures for invariant partitions of a controlled invariant set respectively. We mainly show that Bowen and packing invariance pressures can be determined via the local lower and upper invariance pressures of probability measures, which are analogues of Billingsley's Theorem for the Hausdorff dimension; and give variational principles between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures under some technical assumptions.
In this paper we investigate conditions for maximal regularity of Volterra equations defined on the Lebesgue space of sequences
We study integrability –in the sense of admitting recursion operators– of two nonlinear equations which are known to possess compacton solutions: the
and the CSS equation introduced by Coooper, Shepard, and Sodano,
We obtain a full classification of integrable
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in
In this paper, we consider the augmented Hessian equations
We prove weighted
This paper is concerned with the initial-boundary value problem for the classical Keller–Segel system
in a bounded domain
and
problem (1) possesses a unique global classical solution which is bounded and converges to the trivial state
We consider a nonlocal family of Gross–Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.
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