Small Data Solutions Research Articles

On the Cauchy problem for a weakly coupled system of semi-linear σ-evolution equations with double dissipation

In this paper, we would like to consider the Cauchy problem for a multi-component weakly coupled system of semi-linear σ-evolution equations with double dissipation for any σ≥1. The first main purpose is to obtain the global (in time) existence of small data solutions in the supercritical condition by assuming additional L1 regularity for the initial data and using multi-loss of decay wisely. For the second main one, we are interested in establishing the blow-up results together with sharp estimates for lifespan of solutions in the subcritical case. The proof is based on a contradiction argument with the help of modified test functions to derive the upper bound estimates. Finally, we succeed in catching the lower bound estimate by constructing Sobolev spaces with the time-dependent weighted functions of polynomial type in their corresponding norms.

Modified scattering for the small data solutions to the Vlasov–Maxwell system

Modified scattering for the small data solutions to the Vlasov–Maxwell system

Global Existence of Small Data Solutions to Weakly Coupled Systems of Semi-Linear Fractional σ–Evolution Equations with Mass and Different Nonlinear Memory terms

We study in this paper the long-term existence of solutions to the system of weakly coupled equations with fractional evolution and various nonlinearities. Our objective is to determine the connection between the regularity assumptions on the initial data, the memory terms, and the permissible range of exponents in a specific equation . Using Lp−Lq estimates for solutions to the corresponding linear fractional σ–evolution equations with vanishing right-hand sides, and applying a fixed-point argument, the existence of small data solutions is established for some admissible range of powers (p1,p2,…,pk).

Dispersive decay bound of small data solutions to Kawahara equation in a finite time scale

In this article, we prove that small localized data yield solutions to Kawahara-type equations which have linear dispersive decay on a finite time scale depending on the size of the initial data. We use the similar method used by Ifrim and Tataru to derive the dispersive decay bound of the solutions to the KdV equation, with some steps being simpler. This result is expected to be the first result of the small data global bounds of the fifth-order dispersive equations with quadratic nonlinearity.

Small Data Solutions for the Vlasov–Poisson System with a Repulsive Potential

Small Data Solutions for the Vlasov–Poisson System with a Repulsive Potential

Dispersive decay of small data solutions for the KdV equation

Dispersive decay of small data solutions for the KdV equation

On small-data solution of the chemotaxis–SIS epidemic system with bilinear incidence rate

This paper deals with the initial–boundary value problem for chemotaxis susceptible-infected-susceptible epidemic system with bilinear incidence rate ut=d1Δu+χ∇⋅(u∇v)−β(x)uv+γ(x)v,x∈Ω,t>0,vt=d2Δv+β(x)uv−γ(x)v,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥1), where d1>0, d2>0, χ∈R, 0<β∈C1(Ω¯) and 0<γ∈C1(Ω¯). It is proved that if (u(x,0),v(x,0)) in L∞(Ω)×L1(Ω) is suitably small and ‖v(x,0)‖L∞(Ω)+‖∇v(x,0)‖Lq(Ω)≤K for each K>0 and q>n, then the problem possesses a unique global classical solution which is bounded in Ω×(0,∞). Moreover, we prove that the solution exponentially stabilizes to the disease-free equilibrium N|Ω|,0 with N=∫Ω(u(x,0)+v(x,0)) in L∞(Ω) as t→∞.

Dispersive decay bound of small data solutions to higher order KdV-type equations with low order nonlinearity

Dispersive decay bound of small data solutions to higher order KdV-type equations with low order nonlinearity

On the Cauchy problem for semilinear σ‐evolution equations with time‐dependent damping

In this paper, we would like to consider the Cauchy problem for semilinear ‐evolution equations with time‐dependent damping for any . Motivated strongly by the classification of damping terms in some previous papers, the first main goal of the present work is to make some generalizations from to and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so‐called effective damping cases. For the next main goals, we are going not only to prove the global well‐posedness property of small data solutions but also to indicate blow‐up results for solutions to the semilinear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow‐up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results.

Local/global existence analysis of fractional wave equations with exponential nonlinearity

In this paper, we concern with an exponential nonlinearity for a fractional wave equation in the whole space, we establish the local existence of solutions in a dense subspace of the Orlicz classification. Moreover, we obtain the global existence of solutions for small initial data in lower dimension 1≤d≤3. Our proofs base on the analyticity of Mittag-Leffler functions, the framework of prior estimates and the type of exponential nonlinearity.

Global behavior of small data solutions for the 2D Dirac–Klein-Gordon system

In this paper, we are interested in the two-dimensional Dirac–Klein-Gordon system, which is a basic model in particle physics. We investigate the global behavior of small data solutions to this system in the case of a massive scalar field and a massless Dirac field. More precisely, our main result is twofold: (1) we show sharp time decay for the pointwise estimates of the solutions, which implies the asymptotic stability of this system; (2) we show the linear scattering result of this system which is a fundamental problem when it is viewed as dispersive equations. Our result is valid for general small, high-regular initial data, and in particular, there is no restriction on the support of the initial data.

On the Role of the Displacement Current and the Cattaneo’s Law on Boundary Layers of Plasma

In the present paper, we aim to mathematically analyse the role of the displacement current and the Cattaneo’s law on the boundary-layer theory of plasma, when the corresponding characteristic speed is relativistic. We restrict our analysis to two-dimensional flows and we study the asymptotic limit of the Navier–Stokes–Maxwell equations with Cattaneo’s law near a bounding flat line, when the Hartmann, Reynolds and magnetic Reynolds numbers proportionally diverge to infty . The goal of this paper is twofold. We first show that the extended version of the Navier–Stokes–Maxwell equations leads to a new family of boundary layers, which are hyperbolic both on the momentum equation and the Ampere’s law. Secondly, we address the well-posedness of the derived equations and show the existence of global-in-time analytic solutions for small initial data. Our modelling highlights which conditions on the dimensionless parameters allow to interpret the proposed system as boundary layers with thickness typical of Prandtl or Hartmann. Furthermore, our development shows that the conditions related to Hartmann might be more physically acceptable. Finally, our analysis suggests that the Cattaneo’s law and the displacement current might indeed stabilize the derived system in terms of existence of global-in-time analytic solutions.

Global classical solutions to equatorial shallow-water equations

In this article, we study the equatorial shallow-water equations with slip boundary condition in bounded domain. By exploring the dissipative structures of the system, we obtaining a priori estimates of the solution for small initial data. Then the existence of classical global solutions and exponential stability results are given.&#x0D; For more inofrmation see https://ejde.math.txstate.edu/Volumes/2023/62/abstr.html

Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up

We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space : \begin{align} \left\{\begin{array}{ll} \partial_{t}u=\Delta_{\mathbb{H}^{n}} u+ f(u, t) & \hbox{ in }~ \mathbb{H}^{n}\times (0, T), \\ \quad u =u_{0} & \hbox{ in }~ \mathbb{H}^{n}\times \{0\}. \end{array}\right. \tag{0.1} \end{align} We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, (0.1) exhibits Fujita phenomena for the exponential weight $h(t) = e^{\mu t},$ i.e., there exists a critical exponent $ \mu^*$ such that if $\mu > \mu^*$, then all non-negative solutions blow-up in finite time and if $\mu \leq \mu^*$, there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that (0.1) with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.

Nonlinear fractional damped wave equation on compact Lie groups

In this paper, we deal with the initial value fractional damped wave equation on G, a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on G. Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, i.e., damping and mass with the same power type nonlinearity on compact Lie groups, and prove the global in-time existence of small data solutions in the energy evolution space.

The σ$$ \sigma $$‐evolution equations with friction and unbounded coefficients

We present in this article the study on the decay estimates for solutions of the Cauchy problem for a class of linear ‐evolution equations with unbounded coefficients with in space dimension , where the coefficient is continuous and unbounded satisfying the asymptotic behavior as , with the parameter , and is a positive and bounded continuous function. By investigating the diffusion phenomenon, we obtain new results that generalize our previous study for the damped wave models with constant and nearly constant coefficients. Using the decay estimates for the solution of the linear problem, we establish a result on the global (in time) existence of the small data solution for the corresponding semilinear Cauchy problem with the nonlinearity of power type.

Weakly Coupled System of Semi-Linear Fractional θ-Evolution Equations with Special Cauchy Conditions

In this paper, we consider a weakly system of fractional θ-evolution equations. Using the fixed-point theorem, a global-in-time existence of small data solutions to the Cauchy problem is proved for one single equation. Using these results, we prove the global existence for the system under some mixed symmetrical conditions that describe the interaction between the equations of the system.

Global strong solution of 3D inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity

In this paper, we investigate the Cauchy problem of the three-dimensional inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity. For the case μ(ρ)=ρ, we prove the global existence of strong solution for small initial data satisfying the compatibility condition. Furthermore, we also obtain the large time algebraic decay rates of the velocity. The innovation of this article is that it does not require a positive lower bound for viscosity.

A note to semilinear de Sitter models in 1d with balanced mass and dissipation

In this paper we consider the Cauchy problem for semilinear de Sitter models in 1d with balanced mass and dissipation. The model of interest is ϕtt−e−2tϕxx+ϕt+14ϕ=|ϕ|p,(ϕ(0,x),ϕt(0,x))=(0,g(x)).We study the global (in time) existence of small data solutions. In particular, we show by applying Schauder’s fixed point theorem that there exists a Sobolev solution for all p>1.

Asymptotic behavior of solutions to coupled semilinear parabolic equations with general degenerate diffusion coefficients

&lt;p style='text-indent:20px;'&gt;This paper concerns the asymptotic behavior of solutions to one-dimensional coupled semilinear degenerate parabolic equations with superlinear reaction terms both in bounded and unbounded intervals. The equations are degenerate at a lateral boundary point and the diffusion coefficients are general functions. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large ones in the case that the degeneracy is not strong. Whereas in the case that the degeneracy is strong enough, the nontrivial solution must blow up in a finite time. For the problem in an unbounded interval, blowing-up theorems of Fujita type are established. If the degeneracy of the equations at the lateral boundary point is strong enough, it is shown that any nontrivial solution must blow up in a finite time. If the degeneracy of the equations at the lateral boundary point is not strong, it is proved that the critical Fujita curve is determined by the asymptotic behavior of the diffusion coefficient at infinity. Furthermore, the critical case is also considered.&lt;/p&gt;