Existence Of Global Solutions Research Articles

Asymptotics of Solutions for Periodic Problem for the Korteweg-de Vries Equation with Landau Damping, Pumping and Higher Order Convective Non Linearity

We study the periodic problem for the Korteweg–de Vries equation with Landau damping, linear pumping and a higher-order convective nonlinearity wt+wxxx-αwxx=βw+λwx2wxx,x∈Ω,t>0,w(0,x)=ψx,x∈Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{c} w_{t}+w_{xxx}-\\alpha w_{xx}=\\beta w+\\lambda w_{x}^{2}w_{xx},\ ext { }x\\in \\Omega ,t>0,\\\\ w(0,x)=\\psi \\left( x\\right) ,\ ext { }x\\in \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where, alpha ,beta >0,lambda in mathbb {R},Omega =left[ -pi ,pi right] . We assume that the initial data psi left( xright) are 2pi - periodic. We prove the global existence of solutions and analyze their large-time asymptotics.

Mean‐square exponential input‐to‐state stability of stochastic delayed recurrent neural networks with local Lipschitz condition

We consider the global existence of solutions and input‐to‐state stability for a class of stochastic delayed recurrent neural networks without uniform Lipschitz condition. Under local Lipschitz condition, we find new sufficient conditions that ensure the solutions of given neural networks exist globally and are mean‐square exponentially input‐to‐state stable. Furthermore, we highlight the advantages of our novel results by comparing with some known results as well as a numerical example.

Large-time behavior of global solutions to the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas

This paper is concerned with the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas when the viscosity is a positive constant or depends on the density. According to Saha's ionization equation, the equations of state for the basic thermodynamic variables of an ionized gas depend also on the degree of ionization, which leads to the loss of concavity of the physical entropy in some small bounded domain. Such a property of the ionized gas together with the unboundedness of the domain under our consideration makes it hard to derive the desired dissipative estimates on the first-order spatial derivative of both the bulk velocity and the absolute temperature, thus making the problem more challenging.In this paper, for a class of constant non-vacuum equilibrium states with positive temperature, we succeed in deducing the desired uniform-in-time bounds on the above mentioned dissipative estimates on both the bulk velocity and the absolute temperature. Based on such an estimate, we can then establish the global existence of solutions to the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas with positive constant or density-dependent viscosity and show that such a class of constant non-vacuum equilibrium states is time asymptotically nonlinear stable.

On the Analysis of Regularized Fuzzy Systems of Uncertain Differential Equations.

This article analyzes a regularized set of fuzzy differential equations with respect to an uncertain parameter. We provide sufficient conditions for the correctness of a new regularization scheme. For the resulting family of regularized fuzzy differential equations, the following properties are analyzed, and efficient criteria are proposed: successive approximations, continuity, global existence of solutions, existence of approximate solutions, existence of solutions in the autonomous case. In addition, we develop stability criteria for the regularized family of fuzzy differential equations on the basis of the comparison technique and the method of nonlinear integral inequalities. We expect that the derived results will inspire future research work in this direction.

Global existence, asymptotic stability and blow up of solutions for a nonlinear viscoelastic plate equation involving (p(x), q(x))-Laplacian operator

This study aims at investigating the global existence, asymptotic stability and blow up of solutions for a nonlinear viscoelastic fourth-order (p(x), q(x))-Laplacian equation with variable-exponent nonlinearities. First, we prove the global existence of solutions, and next, we show that the solutions are asymptotically stable if initial data p(x) and q(x) are in the appropriate range. Moreover, under suitable conditions on initial data, we prove that there exists a finite time in which some solutions blow up with positive as well as negative initial energies.

On the Fisher‐KPP model with nonlocal nonlinear sources

AbstractThe Cauchy problem considered in this paper is the following: where . When the coefficient remains positive, (1) is analogous to It is well known that when , the local solution of (2) blows up in finite time as long as the initial value is nontrivial. The main aim of this paper is to obtain sufficient conditions for global existence of solutions to (1) for and thus it forms a contrast to (2). The exponent produces an exact balance in the mass‐invariant scaling of diffusion and growth in (1). In particular, we prove that if , and where is the sharp constant of the inequality , then (1) admits a global classical solution. Moreover, the long time behavior of the solution is also discussed. In addition, we prove that when , the problem is solvable globally in time without any restriction on the initial data.

The Large Deviation of Semi-linear Stochastic Partial Differential Equation Driven by Brownian Sheet

We prove the the large deviation principle(LDP) for the law of the one-dimensional semilinear stochastic partial differential equations driven by nonlinear multiplicative noise. Firstly, combining the energy estimate and approximation procedure, we obtain the existence of global solution. Then the large deviation principle is obtained via weak convergence method.

Quasi self-similarity and its application to the global in time solvability of a superlinear heat equation

This paper concerns the global in time existence of solutions for a semilinear heat equation (P)∂tu=Δu+f(u),x∈RN,t>0,u(x,0)=u0(x)≥0,x∈RN,where N≥1, u0 is a nonnegative initial function and f∈C1([0,∞))∩C2((0,∞)) denotes superlinear nonlinearity of the problem. We consider the global in time existence and nonexistence of solutions for problem (P). The main purpose of this paper is to determine the critical decay rate of initial functions for the global existence of solutions. In particular, we show that it is characterized by quasi self-similar solutions which are solutions W of ΔW+y2⋅∇W+f(W)F(W)+f(W)+|∇W|2f(W)F(W)[q−f′(W)F(W)]=0in y∈RN, where F(s)≔∫s∞1f(η)dη and q≥1.

Exponential Stability for Damped Shear Beam Model and New Facts Related to the Classical Timoshenko System With a Distributed Delay Term

We consider in this manuscript  a Timoshenko type beam model with a distributed delay term.  If the distributed delay term is small enough, we prove the global existence of solutions by using the Faedo-Galerkin approximations together with some energy estimates. Under suitable assumptions, we prove exponential stability of the solution. This result is obtained by introducing a suitable Lyapounov function.

Forces for the Navier–Stokes Equations and the Koch and Tataru Theorem

We consider the Cauchy problem for the incompressible Navier--Stokes equations on the whole space $\mathbb{R}^3$, with initial value $\vec u_0\in {\rm BMO}^{-1}$ (as in Koch and Tataru's theorem) and with force $\vec f=\Div \mathbb{F}$ where smallness of $\mathbb{F}$ ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin's solutions (in $L^2\mathcal{F}^{-1}L^1$) or solutions in the multiplier space $\mathcal{M}(\dot H^{1/2,1}_{t,x}\mapsto L^2_{t,x})$.

On the asymptotic behavior of solutions to a class of grand canonical master equations

In this article, we investigate the long-time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential.We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitably weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly but not all. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature.

Qualitative properties of solution to a viscoelastic Kirchhoff-like plate equation

This paper is concerned with the initial boundary value problem for viscoelastic Kirchhoff-like plate equations with rotational inertia, memory, p-Laplacian restoring force, weak damping, strong damping, and nonlinear source terms. We establish the local existence and uniqueness of the solution by linearization and the contraction mapping principle. Then, we obtain the global existence of solutions with subcritical and critical initial energy by applying potential well theory. Then, we prove the asymptotic behavior of the global solution with positive initial energy strictly below the depth of the potential well. Finally, we conduct a comprehensive study on the finite time blow-up of solutions with negative initial energy, null initial energy, and positive initial energy strictly below the depth of the potential well and arbitrary positive initial energy, respectively.

Existence and Qualitative Properties of Solution for a Class of Nonlinear Wave Equations with Delay Term and Variable-Exponents Nonlinearities

This article is devoted to a study of the question of existence (in time) of weak solutions and the derivation of qualitative properties of such solutions for the nonlinear viscoelastic wave equation with variable exponents and minor damping terms. By using the energy method combined with the Faedo–Galerkin method, the local and global existence of solutions are established. Then, the stability estimate of the solution is obtained by introducing a suitable Lyapunov function.

Existence and Decay of Solutions for a Parabolic-Type Kirchhoff Equation with Variable Exponents

This paper deals with a parabolic-type Kirchhoff equation with variable exponents. Firstly, we obtain the global existence of solutions by Faedo-Galerkin method. Later, we prove the decay of solutions by Komornik's inequality.

The Fokas-Lenells equation on the line: Global well-posedness with solitons

In this paper, we prove the existence of global solutions in H3(R)∩H2,1(R) to the Fokas-Lenells (FL) equation on the line when the initial data includes solitons. A key tool in proving this result is a newly modified Darboux transformation, which adds or subtracts a soliton with given spectral and scattering parameters. In this way the inverse scattering transform technique is then applied to establish the global well-posedness of initial value problem with a finite number of solitons based on our previous results on the global well-posedness of the FL equation.

Well-posedness and exponential stability for the logarithmic Lamé system with a time delay

ABSTRACT This paper is concerned with the initial-boundary value problem for a logarithmic Lamé system with a time delay in a bounded domain. We prove the well-posedness of the system by utilizing the semigroup theory. Then, we prove the existence of global solutions by using the well-depth method. In addition, we establish an exponential stability decay result under appropriate assumptions on the weight of the time delay and that of frictional damping.

Fisher–KPP equation on the Heisenberg group

AbstractIn this paper, we consider the Fisher–KPP equation on the Heisenberg group. We discuss the existence of global solutions, asymptotic behavior of global solutions and blow‐up solutions. Moreover, we extend the obtained results to the time‐fractional Fisher–KPP equation on the Heisenberg group.

Global existence of solutions to a full parabolic attraction-repulsion chemotaxis fluid system

Global existence of solutions to a full parabolic attraction-repulsion chemotaxis fluid system

Global Dynamics of a Diffusive Within-Host HTLV/HIV Co-Infection Model with Latency

In several publications, the dynamical system of HIV and HTLV mono-infections taking into account diffusion, as well as latently infected cells in cellular transmission has been mathematically analyzed. However, no work has been conducted on HTLV/HIV co-infection dynamics taking both factors into consideration. In this paper, a partial differential equations (PDEs) model of HTLV/HIV dual infection was developed and analyzed, considering the cells’ and viruses’ spatial mobility. CD4+T cells are the primary target of both HTLV and HIV. For HIV, there are three routes of transmission: free-to-cell (FTC), latent infected-to-cell (ITC), and active ITC. In contrast, HTLV transmits horizontally through ITC contact and vertically through the mitosis of active HTLV-infected cells. In the beginning, the well-posedness of the model was investigated by proving the existence of global solutions and the boundedness. Eight threshold parameters that determine the existence and stability of the eight equilibria of the model were obtained. Lyapunov functions together with the Lyapunov–LaSalle asymptotic stability theorem were used to investigate the global stability of all equilibria. Finally, the theoretical results were verified utilizing numerical simulations.

Convergence to a diffusive contact wave for solutions to a system of hyperbolic balance laws

We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.