Small Data Solutions Research Articles

Asymptotic Properties of the Solutions to the Vlasov–Maxwell System in the Exterior of a Light Cone

Abstract This paper is concerned with the asymptotic behavior of small data solutions to the three-dimensional Vlasov–Maxwell system in the exterior of a light cone. The plasma does not have to be neutral, and no compact support assumptions are required on the data. In particular, the initial decay in the velocity variable of the particle density is optimal and we only require an $L^2$ bound on the electromagnetic field with no additional weight. We use vector field methods to derive improved decay estimates in null directions for the electromagnetic field, the particle density, and their derivatives. In contrast with [ 2], where we studied the behavior of the solutions in the whole spacetime, the initial data have less decay and we do not need to modify the commutation vector fields of the relativistic transport operator. To control the solutions under these assumptions, we crucially use the strong decay satisfied by the particle density in the exterior of the light cone, null properties of the Vlasov equation, and certain hierarchies in the energy norms.

A structurally damped σ‐evolution equation with nonlinear memory

In this paper, we investigate the global (in time) existence of small data solutions to the Cauchy problem for the following structurally damped σ‐evolution model with nonlinear memory term: with σ>0. In particular, for γ∈((n−σ)/n,1), we find the sharp critical exponent, under the assumption of small data in L1. Dropping the L1 smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration 1−γ in the nonlinear memory term and the assumption that initial data are small in Lm, for some m>1.

Decay of regular solutions for the critical 2D Zakharov–Kuznetsov equation posed on rectangles

An initial-boundary value problem for the critical generalized 2D Zakharov–Kuznetsov equation posed on rectangles is considered. The existence, uniqueness, and exponential decay rate of global regular solutions for small initial data are established.

Influence of strong time-dependent oscillations in semilinear damped wave models

We study the global (in time) existence of small data solutions to some Cauchy problems for semilinear damped wave models with strong time-dependent oscillations. The goal is to understand the influence of strong oscillations in the coefficients on solutions to some semilinear models with an “effective-like” damping term, where the data are supposed to belong to different classes of regularity.

Sharp Asymptotic Behavior of Solutions of the 3d Vlasov\u2013Maxwell System with Small Data

We study the asymptotic properties of the small data solutions of the Vlasov–Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For the Vlasov field and its derivatives, we obtain, as in Fajman et al. (The Stability of the Minkowski space for the Einstein-Vlasov system, 2017. arXiv:1707.06141), optimal pointwise decay estimates by a vector field method where the commutators are modification of those of the free relativistic transport equation. In order to control high velocities and to deal with non integrable source terms, we make fundamental use of the null structure of the system and of several hierarchies in the commuted equations.

On one-dimension quasilinear wave equations with null conditions

In this paper, we show that one-dimension systems of quasilinear wave equations with null conditions admit global classical solutions for small initial data. This result extends Luli, Yang and Yu's seminal work [G. Luli, S. Yang, P. Yu, On one-dimension semi-linear wave equations with null conditions, Adv. Math.329 (2018) 174-188] from the semilinear case to the quasilinear case. Furthermore, we also prove that the global solution is asymptotically free in the energy sense. In order to achieve these goals, we will employ Luli, Yang and Yu's weighted energy estimates with positive weights, introduce some space-time weighted energy estimates and pay some special attentions to the highest order energies, then use some suitable bootstrap process to close the argument.

Global existence and blow-up of solutions of the time-fractional space-involution reaction-diffusion equation

A time-fractional space-nonlocal reaction-diffusion equation in a bounded domain is considered. First, the existence of a unique local mild solution is proved. Applying Poincaré inequality it is obtained the existence and boundedness of global classical solution for small initial data. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time.

Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion

The global small solutions of the tropical climate model are obtained with the fractional dissipative terms Λαu in the equation of the barotropic mode u and Λαv in the equation of the first baroclinic mode v. More precisely, we prove for 1<α ≤ 2 that the couple system has global unique strong solutions for small initial data with critical regularities. Moreover, the smallness assumption imposed on the initial barotropic mode of the velocity can be removed if α=2. We also study the large time behavior of the constructed solutions and obtain optimal time decay rates by a pure energy argument.

Existence of global solutions for a weakly coupled system of semilinear viscoelastic damped $\sigma$-evolution equations

We prove the global (in time) existence of small data solutions from energy spaces based on L q spaces, q ∈ ( 1 , ∞ ) , to the Cauchy problem for a weakly coupled system of semilinear viscoelastic damped σ -evolution equations, where we consider nonlinearity terms with powers p 1 , p 2 > 1 and any σ 1 , σ 2 ≥ 1 in the comparison between two single equations. To do this, by mixing additional L m regularity for the data on the basis of L q - L q estimates, with q ∈ ( 1 , ∞ ) and m ∈ [ 1 , q ) , we apply ( L m ∩ L q ) - L q estimates for solutions to the corresponding linear Cauchy problems to treat semilinear problems. In addition, two different strategies allowing no loss of decay and some loss of decay combined with the flexible choice of admissible parameters σ 1 , σ 2 , m and q bring some benefits to relax the restrictions on the admissible exponents p 1 , p 2 .

Decay properties and asymptotic profiles for elastic waves with Kelvin–Voigt damping in 2D

In this paper we consider elastic waves with Kelvin-Voigt damping in 2D. For the linear problem, applying pointwise estimates of the partial Fourier transform of solutions in the Fourier space and asymptotic expansions of eigenvalues and their eigenprojections, we obtain sharp energy decay estimates with additional $L^m$ regularity and $L^p-L^q$ estimates on the conjugate line. Furthermore, we derive asymptotic profiles of solutions under different assumptions of initial data. For the semilinear problem, we use the derived $L^2-L^2$ estimates with additional $L^m$ regularity to prove global (in time) existence of small data solutions to the weakly coupled system. Finally, to deal with elastic waves with Kelvin-Voigt damping in 3D, we apply the Helmholtz decomposition.

Global existence and asymptotic behavior of solutions to the Euler equations with time-dependent damping

ABSTRACT We study the isentropic Euler equations with time-dependent damping, given by . Here, λ and μ are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when in multi-dimensions . Our strategy of proving the global existence is to convert the Euler system to a time-dependent damped wave equation and use a kind of weighted energy estimate. Investigation to the asymptotic behavior of the solution is based on the detailed analysis to the fundamental solutions of the corresponding linear damped wave equation and it coincides with that of standard results if λ deduces to zero.

Semilinear wave models with friction and viscoelastic damping

In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear wave equation with friction, viscoelastic damping, and a power nonlinearity. We are interested in the connection between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity.

Asymptotic behaviors of global solutions for a semilinear diffusion equation in the de Sitter spacetime

A semilinear diffusion equation is considered in the de Sitter spacetime with the spatially flat curvature. Global solutions for small initial data and their asymptotic behaviors in high order are obtained. Some effects of the spatial expansion and contraction are studied through the problem.

The global classical solution to compressible Euler system with velocity alignment

In this paper, the compressible Euler system with velocity alignment and damping is considered, where the influence matrix of velocity alignment is not positive definite. Sound speed is used to reformulate the system into symmetric hyperbolic type. The global existence and uniqueness of smooth solution for small initial data is provided.

The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity

We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.

Global Small data Solutions for a system of semilinear heat equations and the corresponding system of damped wave equations with nonlinear memory

We consider the Cauchy problem for a strongly coupled semi-linear heat equations with some kind of nonlinearity in multi-dimensional space ℝN. We see under some conditions on the exponents and on the dimension N, that the existence and uniqueness of time-global solutions for small data and their asymptotic behaviors are obtained. This observation will be applied to the corresponding system of the damped wave equations in low dimensional space.

A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain

We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity $ |u|^p+|u_t|^q $ in an exterior domain, where $ p, q>1 $. We prove global (in time) existence of small data solution with suitable higher regularity by using a weighted energy method, and assuming some conditions on powers of nonlinearity.

Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity

In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power non-linearity. We prove that the critical exponent is the Fujita exponent pFuj(Q)=1+2/Q, where Q is the homogeneous dimension of the Heisenberg group.On the one hand, we will prove the global existence of small data solutions for p>pFuj(Q) in an exponential weighted energy space. On the other hand, a blow-up result for 1<p≤pFuj(Q) under certain integral sign assumptions for the Cauchy data by using the test function method.

Well-posedness of semilinear heat equations in L1

The problem of obtaining necessary and sufficient conditions for local existence of non-negative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term f has only recently been resolved (Laister et al. (2016)). There, for the more difficult case of initial data in L1, a necessary and sufficient integral condition on f emerged. Here, subject to this integral condition, we consider other fundamental properties of solutions with L1 initial data of indefinite sign, namely: uniqueness, regularity, continuous dependence and comparison. We also establish sufficient conditions for the global-in-time continuation of solutions for small initial data in L1.

A Note on $$L^{p}-L^{q}$$ Estimates for Semilinear Critical Dissipative Klein–Gordon Equations

In this note, we consider a semilinear wave equation with scale-invariant mass and dissipation. The scale of the dissipation is the same of the mass term and this creates an interplay in which a relation between the coefficients comes into play to determine if the dissipation is dominant with respect to the mass, or viceversa. We prove global existence of small data solutions for supercritical power nonlinearities when the mass term is dominant with respect to the dissipative term (“Klein–Gordon type” case). The critical exponent is a modified Strauss exponent for global small data solutions to semilinear Klein–Gordon equations.