Nonlinear Differential Equations and Applications (NoDEA)最新文献

We discuss averaging for dispersion-managed nonlinear Schrödinger equations in the fast dispersion management regime,with an application to the problem of constructing soliton-like solutions to dispersion-managed nonlinear Schrödinger equations.

In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by ((-Delta )^{alpha }) and magnetic diffusion given by reducing about the square root of logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power (alpha ) is a positive constant. In addition, we prove several global a priori bounds for the case (alpha =0). Finally, for the case (alpha =0), it is also shown that the control of the direction of the magnetic field in a suitable norm is enough to guarantee the global regularity. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global regularity issue on the 2D resistive MHD equations, namely, the case when the MHD equations only have standard Laplacian magnetic diffusion.

In bounded, spatially two-dimensional domains, the system

complemented with initial and homogeneous Neumann boundary conditions, models the interaction between prey (with density u), predator (with density v) and superpredator (with density w), which preys on both other populations. Apart from random motion and prey-tactical behavior of the primary predator, the key aspect of this system is that the secondary predator reacts to alarm calls of the prey, issued by the latter whenever attacked by the primary predator. We first show in the pure alarm-taxis model, i.e. if (xi = 0), that global classical solutions exist. For the full model (with (xi > 0)), the taxis terms and the presence of the term (-a_2 uw) in the first equation apparently hinder certain bootstrap procedures, meaning that the available regularity information is rather limited. Nonetheless, we are able to obtain global generalized solutions. An important technical challenge is to guarantee strong convergence of (weighted) gradients of the first two solution components in order to conclude that approximate solutions converge to a generalized solution of the limit problem.

We prove the existence of sign-changing solution to the problem

where (mathbb {R}^N_+ = {(x',x_N): x' in mathbb {R}^{N-1},,x_N>0 }) is the upper half-space, (2_*:=2(N-1)/(N-2)), (N ge 7), (frac{partial u}{partial nu }) is the partial outward normal derivative and the parameter (lambda >0) interacts with the spectrum of the linearized problem. In the proof, we apply variational methods.

In this study, we investigate the existence of solutions ((lambda , u) in mathbb {R} times H^1(mathbb {R}^N)) to the Schrödinger equation

where (Nge 2), (a>0) is a constant and p satisfies (2+4/N<p<+infty ). The potential V satisfies the condition that the operator (-Delta +V) contains infinitely many isolated eigenvalues with an accumulation point. We prove that this equation has a sequence of solutions ({(lambda _m, u_m)}) such that (Vert u_mVert _{L^infty (mathbb {R}^N)}rightarrow 0) as (mrightarrow infty ). The proof is provided by establishing a new critical point theorem without the typical Palais–Smale condition.

We establish a convergence theorem for Crandall–Lions viscosity solutions to path-dependent Hamilton–Jacobi–Bellman PDEs. Our proof is based on a novel convergence theorem for dynamic sublinear expectations and the stochastic representation of viscosity solutions as value functions.

We consider a curve with boundary points free to move on a line in ({{{mathbb {R}}}}^2), which evolves by the (L^2)-gradient flow of the elastic energy, that is, a linear combination of the Willmore and the length functional. For this planar evolution problem, we study the short and long-time existence. Once we establish under which boundary conditions the PDE’s system is well-posed (in our case the Navier boundary conditions), employing the Solonnikov theory for linear parabolic systems in Hölder space, we show that there exists a unique flow in a maximal time interval [0, T). Then, using energy methods we prove that the maximal time is (T= + infty ).

In this paper we study Monge solutions to stationary Hamilton–Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result.

In this paper, we study the dynamical behavior of a linear control system on (mathbb {R}^2) when the associated matrix has real eigenvalues. Different from the complex case, we show that the position of the control zero relative to the control range can have a strong interference in such dynamics if the matrix is not invertible. In the invertible case, we explicitly construct the unique control set with a nonempty interior.

We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of [11] in the setting of the standard cubic-quintic NLS.