Journal of Evolution Equations最新文献

We prove that for bounded, divergence-free vector fields in (L^1_textrm{loc}((0,+infty );BV_textrm{loc}(mathbb {R}^d;mathbb {R}^d))), regularisation by convolution of the vector field selects a single solution of the transport equation for any locally integrable initial datum. We recall the vector field constructed by Depauw in (C R Math Acad Sci Paris 337:249–252, 2003), which lies in the above class of vector fields. We show that the transport equation along this vector field has at least two bounded weak solutions for any bounded initial datum.

We consider the barotropic Euler equations with pairwise attractive Riesz interactions and linear velocity damping in the periodic domain. We establish the global-in-time well-posedness theory for the system near an equilibrium state if the coefficient of the Riesz interaction term is small. We also analyze the large-time behavior of solutions showing the exponential rate of convergence toward the equilibrium state as time goes to infinity.

We prove the global existence and uniqueness of solutions to the Vlasov–Riesz–Fokker–Planck system around the global Maxwellian in the periodic spatial domain. Depending on the order of Riesz potential, we present two frameworks for the construction of global-in-time solutions with Sobolev and analytic regularity. The analytic function framework covers the Vlasov–Dirac–Benney–Fokker–Planck system. Furthermore, we show the exponential decay of solutions toward the global Maxwellian. Our result is generalized to the whole space case in which the decay rate of convergence is algebraic.

This paper is concerned with propagation phenomenon of a three species competition system with nonlocal dispersal in shifting habitats. We first give the existence of two types of forced wave connecting origin to only one species state and semi-co-existence state in supercritical and critical cases. Then, we get the existence of forced waves connecting origin to coexistence state at any speed. In particular, we establish the spreading property of the associated Cauchy problem depending on the range of the shifting speed which is identified respectively by (i) extinction of three species; (ii) only one species surviving; (iii) two species coexisting; (iv) persistence of three species.

The aim of this paper is to establish the stability of pullback random attractors of non-autonomous fractional stochastic p-Laplacian equations driven by nonlinear colored noise. In order to overcome the difficulties caused by lack of compact Sobolev embedding on unbounded domains and weak dissipative structure of the equation, we first prove the existence, uniqueness and backward compactness of a special kind of pullback random attractor using the method of spectral decomposition in bounded domains and the uniform tail-estimates of solutions outside bounded domains over the infinite time interval. The measurability of this class of attractors is established by proving that the two classes of defined attractors are equal with respect to two different universes. Finally, the stability of the attractors is investigated by assuming that the time-dependent external forcing term converges to the time-independent external force as the time parameter tends to negative infinity.

In this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is the cubic one convolved with the Coulomb potential (|x|^{-1}), and it produces the long-range interaction in the sense of scattering phenomenon. From this observation, one anticipates that small solutions converge to modified scattering states, although they decay as linear solutions. We show the global well-posedness and the modified scattering for small solutions in weighted Sobolev spaces. Our proof follows a road map of exploiting the space-time resonance by Germain et al. (Int Math Res Not 2009(3):414–432, 2008), and Pusateri (Commun Math Phys 332(3):1203–1234, 2014). Compared to the result in three dimensional case (Pusateri 2014), weaker time decay in two dimension is one of the main obstacles.

In this paper, we consider a dynamic model of fracture for viscoelastic materials, in which the constitutive relation, involving the Cauchy stress and the strain tensors, is given in an implicit nonlinear form. We prove the existence of a solution to the associated viscoelastic dynamic system on a prescribed time-dependent cracked domain via a discretization-in-time argument. Moreover, we show that such a solution satisfies an energy-dissipation balance in which the energy used to increase the crack does not appear. As a consequence, in analogy to the linear case this nonlinear model exhibits the so-called viscoelastic paradox.

We consider differential operators A that can be represented by means of a so-called closure relation in terms of a simpler operator (A_{{text {ext}}}) defined on a larger space. We analyse how the spectral properties of A and (A_{{text {ext}}}) are related and give sufficient conditions for exponential stability of the semigroup generated by A in terms of the semigroup generated by (A_{{text {ext}}}). As applications we study the long-term behaviour of a coupled wave–heat system on an interval, parabolic equations on bounded domains that are coupled by matrix-valued potentials, and of linear infinite-dimensional port-Hamiltonian systems with dissipation on an interval.

In this paper, we consider the inhomogeneous nonlinear Schrödinger equation (ipartial _t u +Delta u =K(x)|u|^alpha u,; u(0)=u_0in H^1({mathbb {R}}^N),; Nge 3,; |K(x)|+|x||nabla K(x)|lesssim |x|^{-b},; 0<b< min (2, N-2),; 0<alpha <{(4-2b)/(N-2)}). We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted (L^2)-space for a new range (alpha _0(b)<alpha <(4-2b)/N). The value (alpha _0(b)) is the positive root of (Nalpha ^2+(N-2+2b)alpha -4+2b=0,) which extends the Strauss exponent known for (b=0). Our results improve the known ones for (K(x)=mu |x|^{-b}), (mu in {mathbb {C}}). For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range of (alpha ). In the defocusing case, we prove decay estimates provided that the potential satisfies some rigidity-type condition which leads to a scattering result. We give also a new scattering criterion taking into account the potential K.

This paper studies regularity properties of the weak solutions to the 3D Navier–Stokes equations with damping in the whole space and bounded domains. We find the space restriction on the initial velocity to guarantee the local existence of strong solutions. Based on it, we complete the existence results for the global strong solutions in the whole space and improve the restriction on the damping exponent for the existence of the global strong solutions in the bounded domains.