Initial Data Decay Research Articles

The coupled Hirota equations with a 3×3$3\times 3$ Lax pair: Painlevé‐type asymptotics in transition zone

AbstractWe consider the Painlevé asymptotics for a solution of the integrable coupled Hirota equations with a Lax pair whose initial data decay rapidly at infinity. Using the Riemann–Hilbert (RH) techniques and Deift–Zhou nonlinear steepest descent arguments, in a transition zone defined by , where is a constant, it turns out that the leading‐order term to the solution can be expressed in terms of the solution of a coupled Painlevé II equations, which are associated with a matrix RH problem and appear in a variety of random matrix models.

The uniform spreading speed in cooperative systems with non-uniform initial data

This paper considers the spreading speed of cooperative nonlocal dispersal systems with irreducible reaction functions and non-uniform initial data. Here the non-uniformity means that all components of initial data decay exponentially but their decay rates are different. It is well-known that in a monostable reaction-diffusion or nonlocal dispersal equation, different decay rates of initial data yield different spreading speeds. In this paper, we show that due to the cooperation and irreducibility of reaction functions, all components of the solution with non-uniform initial data will possess a uniform spreading speed which decreasingly depends only on the smallest decay rate of initial data. The decreasing property of the uniform spreading speed in the smallest decay rate further implies that the component with the smallest decay rate can accelerate the spatial propagation of other components.

On the Cauchy problem of the two-component Novikov-type system with peaked solutions and H1-conservation law

Considered herein is the Cauchy problem for the two-component Novikov-type system with peaked solutions and [Formula: see text]-conservation law. At first, we establish that the solutions maintain corresponding properties at infinity within the lifespan provided that the initial data decay exponentially and algebraically, respectively. Next, the local regularity and analyticity of the solutions to this problem in Sobolev–Gevrey spaces are discussed by a generalized Ovsyannikov theorem in detail.

Effect of decay rates of initial data on the sign of solutions to Cauchy problems of polyharmonic heat equations

In this paper, we consider the sign of solutions to Cauchy problems of linear and semilinear polyharmonic heat equations. Cauchy problems for higher order parabolic equations have no positivity preserving property in general, however, it is expected that solutions to these Cauchy problems are eventually globally positive if initial data decay slowly enough. We first show the existence of the threshold of the decay rate of initial datum which separates whether the corresponding solution to the Cauchy problem of the linear polyharmonic heat equation is eventually globally positive or not. Applying this result, we construct eventually globally positive solutions to the Cauchy problem of the semilinear polyharmonic heat equation under the super-Fujita condition.

On the Cauchy problem for a class of cubic quasilinear shallow-water equations

In this paper, we mainly investigate the Cauchy problem of the cubic Camassa-Holm-type equations which arise as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves, which admit the single peaked solitons and multi-peakon solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model. Then, by overcoming the difficulties cased by the complicated mixed nonlinear structure, we present a precise blow-up scenario. Moreover, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions for Cauchy problem in the Gevrey-Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. Finally, we investigate the persistence properties of the solution to the Cauchy problem, we prove that the solution maintains the corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

Time-dependent incompressible viscous flows around a rigid body: Estimates of spatial decay independent of boundary conditions

We consider the incompressible time-dependent Navier-Stokes system with Oseen term and terms arising in stability problems, in a 3D exterior domain. No boundary conditions are imposed. We consider L2-strong solutions, that is, the velocity u is an L∞-function in time and Lκ-integrable in space for some κ∈[1,3) and some κ∈(3,∞), the spatial gradient ∇xu is L2-integrable in space and in time, and the nonlinearity (u⋅∇x)u is L2-integrable in time and L3/2-integrable in space. It is shown that if the right-hand side of the equation and the initial data decay pointwise in space sufficiently fast, then u and ∇xu also decay pointwise in space, with rates which are higher than those provided by previous theories.

Energy distribution of solutions to defocusing semi-linear wave equation in two dimensional space

We consider finite-energy solutions to the defocusing nonlinear wave equation in two dimensional space. We prove that almost all energy moves to the infinity at almost the light speed as time tends to infinity. In addition, the inward/outward part of energy gradually vanishes as time tends to positive/negative infinity. These behaviours resemble those of free waves. We also prove some decay estimates of the solutions if the initial data decay at a certain rate as the spatial variable tends to infinity. As an application, we prove a couple of scattering results for solutions whose initial data are in a weighted energy space. Our assumption on decay rate of initial data is weaker than previous known scattering results.

Traveling wave solutions of a singular Keller-Segel system with logistic source.

This paper is concerned with the traveling wave solutions of a singular Keller-Segel system modeling chemotactic movement of biological species with logistic growth. We first show the existence of traveling wave solutions with zero chemical diffusion in $ \mathbb{R} $. We then show the existence of traveling wave solutions with small chemical diffusion by the geometric singular perturbation theory and establish the zero diffusion limit of traveling wave solutions. Furthermore, we show that the traveling wave solutions are linearly unstable in the Sobolev space $ H^1(\mathbb{R}) \times H^2(\mathbb{R}) $ by the spectral analysis. Finally we use numerical simulations to illustrate the stabilization of traveling wave profiles with fast decay initial data and numerically demonstrate the effect of system parameters on the wave propagation dynamics.

Admissible Boundary Values for the Gerdjikov-Ivanov Equation with Asymptotically Time-Periodic Boundary Data

We consider the Gerdjikov-Ivanov equation in the quarter plane with Dirichlet boundary data and Neumann value converging to single exponentials $\alpha {\rm e}^{{\rm i}\omega t}$ and $c{\rm e}^{{\rm i}\omega t}$ as $t\to\infty$, respectively. Under the assumption that the initial data decay as $x\to\infty$, we derive necessary conditions on the parameters $\alpha$, $\omega$, $c$ for the existence of a solution of the corresponding initial boundary value problem.

On the Cauchy problem of a new integrable two-component Novikov equation

This paper is devoted to a new integrable two-component Novikov equation with Lax pairs and bi-Hamiltonian structures. Ons the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. On the other hand, we prove that the strong solutions maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self‐similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.

On the Cauchy problem for the new integrable two-component Novikov equation

This paper is devoted to a new integrable two-component Novikov equation, lax pairs and bi-Hamiltonian structures. Firstly, the local well-posedness in nonhomogeneous Besov spaces is established by using the Littlewood–Paley theory and transport equations theory. Then, we verify the blow-up that occurs for this system only in the form of breaking waves. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

Persistence properties and asymptotic behavior for a two-component [formula omitted]-family system with high order nonlinearity

This paper is devoted to the study of persistence properties and asymptotic behavior for a two-component b-family system with high order nonlinearity. We prove that both the density and momentum components of the corresponding solutions with initial compact support will retain the property of being compactly supported throughout its evolution. Moreover, we investigate the asymptotic behaviors of the solutions at infinity within its lifespan when the initial data decay exponentially and algebraically, respectively.

Dynamics of interfaces in the Fisher-KPP equation for slowly decaying initial data

This paper is concerned with the behavior of solutions of the Fisher-KPP equation when initial data decay slowly in space. First, we show by a comparison technique that the motion of an interface (or a thin transition layer) can be approximated as a level set of a first-order PDE of Hamilton-Jacobi type. Then by the method of characteristics, it is shown that various (but not all) motion of interfaces can be observed by taking initial data appropriately. Finally, we study the large-time behavior of solutions, especially, the existence of similarly expanding interfaces.

Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data

AbstractWe consider the Cauchy problem $$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill & {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill & {x\in {\bf R}^N,} \hfill \cr } } \right.$$where N > 2, p > 1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x| → ∞, and investigate the convergence property of the global solutions to the forward self-similar solutions. We first give the precise description of the relationship between the spatial decay of initial data and the large time behaviour of solutions, and then we show the existence of solutions with a time decay rate slower than the one of self-similar solutions. We also show the existence of solutions that behave in a complicated manner.

On global attractors and radiation damping for nonrelativistic particle coupled to scalar field

We consider the Hamiltonian system of scalar wave field and a single nonrelativistic particle coupled in a translation invariant manner. The particle is also subject to a confining external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. We prove that solutions of finite energy converge, in suitable local energy seminorms, to the set ${\cal S}$ of all stationary states in the long time limit $t\to\pm\infty$. Further we show that the rate of relaxation to a stable stationary state is determined by spatial decay of initial data. The convergence is followed by the radiation of the dispersion wave which is a solution to the free wave equation. Similar relaxation has been proved previously for the case of relativistic particle when the speed of the particle is less than the speed of light. Now we extend these results to nonrelativistic particle with arbitrary superlight velocity. However, we restrict ourselves by the plane particle trajectories. The extension to general case remains an open problem.

On continuity equations in space-time domains

In this paper we consider a class of continuity equations that are conditioned to stay in general space-time domains, which is formulated as a continuum limit of interacting particle systems. Firstly, we study the well-posedness of the solutions and provide examples illustrating that the stability of solutions is strongly related to the decay of initial data at infinity. In the second part, we consider the vanishing viscosity approximation of the system, given with the co-normal boundary data. If the domain is spatially convex, the limit coincides with the solution of our original system, giving another interpretation to the equation.

On a new two-component $b$-family peakon system with cubic nonlinearity

In this paper, we propose a two-component \begin{document}$b$\end{document} -family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component \begin{document}$b$\end{document} -family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan provided the initial data decay exponentially and algebraically.

Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system

In this paper, we study symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Lie symmetry analysis and similarity reductions are performed, some invariant solutions are also discussed. Then prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. Furthermore, we show that the system exhibits unique continuation if the initial momentum \begin{document}$m_0$\end{document} and \begin{document}$n_0$\end{document} are positive.

Blow-up phenomena and persistence properties for an integrable two-component peakon system

In this paper, we are concerned with an integrable two-component peakon system, which was proposed by Xia, Qiao and Zhou. We present a precise blow-up scenario and a new blow-up result for strong solutions to the system. Moreover, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.