Journal of Evolution Equations最新文献

We consider persistence properties of solutions for a generalised wave equation including vibration in elastic rods and shallow water models, such as the BBM, the Dai’s, the Camassa–Holm, and the Dullin–Gottwald–Holm equations, as well as some recent shallow water equations with Coriolis effect. We establish unique continuation results and exhibit asymptotic profiles for the solutions of the general class considered. From these results we prove the non-existence of non-trivial spatially compactly supported solutions for the equation. As an aftermath, we study the equations earlier mentioned in light of our results for the general class.

The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital (nonlinear) instability in appropriate periodic Sobolev spaces. The analysis is based on the well-posedness theory, the smoothness of the data-solution map, and an abstract result of instability of equilibria under nonlinear iterations. The resulting instability criterion is applied to two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a local Hopf bifurcation around a critical value of the velocity. The second family comprises arbitrarily large period waves which arise from a homoclinic (global) bifurcation and tend to a limiting traveling pulse when their fundamental period tends to infinity. In the case of both families, the criterion is applied to conclude their orbital instability under the flow of the nonlinear viscous balance law in periodic Sobolev spaces with same period as the fundamental period of the wave.

We consider the controllability of a class of systems of n Stokes equations, coupled through terms of order zero and controlled by m distributed controls. Our main result states that such a system is null-controllable if and only if a Kalman type condition is satisfied. This generalizes the case of finite-dimensional systems and the case of systems of coupled linear heat equations. The proof of the main result relies on the use of the Kalman operator introduced in [1] and on a Carleman estimate for a cascade type system of Stokes equations. Using a fixed-point argument, we also obtain that if the Kalman condition is verified, then the corresponding system of Navier–Stokes equations is locally null-controllable.

In this paper, we consider the initial value problem of the incompressible Hall-MHD system and prove the global well-posedness in the scaling critical class ({dot{B}}_{p,infty }^{-1+frac{3}{p}}(mathbb {R}^3)times ({dot{B}}_{p,infty }^{-1+frac{3}{p}}(mathbb {R}^3) cap L^{infty }(mathbb {R}^3))) for (3< p < infty ). Moreover, we also refine the smallness conditions and show that our global well-posedness holds for initial data whose ({dot{B}}_{p,infty }^{-1+frac{3}{p}}(mathbb {R}^3))-norm is large, provided that some weaker norm is sufficiently small.

In this paper, we propose the compact convergence approach to deal with the continuity of attractors of some reaction–diffusion equations under smooth perturbations of the domain subject to nonlinear Neumann boundary conditions. We define a family of invertible linear operators to compare the dynamics of perturbed and unperturbed problems in the same phase space. All continuity arising from small smooth perturbations will be estimated by a rate of convergence given by the domain variation in a ({mathcal {C}}^1) topology.

The solvability for infinite-dimensional differential algebraic equations possessing a resolvent index and a Weierstraß form is studied. In particular, the concept of integrated semigroups is used to determine a subset on which solutions exist and are unique. This information is later used for a important class of systems, namely, port-Hamiltonian differential algebraic equations.

In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation

subject to initial boundary value conditions

We derive the intermediate extinction profile with refined asymptotics, and with extinction time T and extinction point 0, the gradient profile behaves as (xrightarrow 0) like

agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions.

We study the Cauchy problem for the advection–diffusion equation (partial _t u + {{,mathrm{textrm{div}},}}(uvarvec{b}) = Delta u) associated with a merely integrable divergence-free vector field (varvec{b}) defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.

In this paper we prove the asymptotic behavior, as t tends to zero, of solutions of nonlinear parabolic equations with initial data belonging to Marcinkiewicz spaces. Namely, that if the initial datum (u_{0}) belongs to (M^{m}(Omega )), then

thus extending to Marcinkiewicz spaces the results which hold for data in Lebesgue spaces.

An evidence of temporal discontinuity of the solution in (F^s_{1, infty }(mathbb {R}^d)) is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in related spaces are also discussed.