Uniqueness Of Global Mild Solution Research Articles

On the existence theory for nonlinear plate equations

This paper is devoted to the theoretical analysis of the nonlinear plate equations in $\mathbb{R}^{n}\times (0,\infty),$ $n\geq1,$ with nonlinearity involving a type polynomial behavior. We prove the existence and uniqueness of global mild solutions for small initial data in $L^{1}(\mathbb{R}^{n})\cap H^s(\mathbb{R}^{n})$-spaces. We also prove the existence and uniqueness of local and global solutions in the framework of Bessel-potential spaces $H^s_p(\mathbb{R}^n)=(I-\Delta)^{s/2}L^p(\mathbb{R}^n).$ In order to derive the existence results we develop new time decay estimates of the solution of the corresponding linear problem.

On a generalized stochastic Burgers' equation perturbed by Volterra noise

In this article, we investigate the existence and uniqueness of local mild solutions for the one-dimensional generalized stochastic Burgers' equation (GSBE) containing a nonlinearity of polynomial type and perturbed by α-regular cylindrical Volterra process and having Dirichlet boundary conditions. The Banach fixed point theorem (or contraction mapping principle) is used to obtain the local solvability results. The L∞-estimate on both time and space for the stochastic convolution involving the α-regular cylindrical Volterra process is obtained with the help of Garsia-Rodemich-Rumsey inequality. Further, the existence and uniqueness of global mild solution of GSBE up to third order nonlinearity is also shown.

Mild Solutions for the Stochastic Generalized Burgers–Huxley Equation

In this work, we consider the stochastic generalized Burgers–Huxley equation perturbed by space–time white noise and discuss the global solvability results. We show the existence of a unique global mild solution to such equation using a fixed point method and stopping time arguments. The existence of a local mild solution (up to a stopping time) is proved via contraction mapping principle. Then, establishing a uniform bound for the solution, we show the existence and uniqueness of global mild solution to the stochastic generalized Burgers–Huxley equation. Finally, we discuss the inviscid limit of the stochastic Burgers–Huxley equation to the stochastic Burgers as well as Huxley equations.

Existence Results for Fractional Semilinear Integrodifferential Equations of Mixed Type with Delay

In this paper, we discuss a class of fractional semilinear integrodifferential equations of mixed type with delay. Based on the theories of resolvent operators, the measure of noncompactness, and the fixed point theorems, we establish the existence and uniqueness of global mild solutions for the equations. An example is provided to illustrate the application of our main results.

WELL-POSEDNESS AND CONVERGENCE FOR TIME-SPACE FRACTIONAL STOCHASTIC SCHRÖGER-BBM EQUATION

<abstract> In this paper, the Banach fixed point theorem combined with Mittag-Leffler functions has been used to obtain the existence and uniqueness of global mild solution for a kind of time-space fractional stochastic Schr򤩮ger-BBM equation driven by Gaussian noise. The spatial-temporal regularity of the nonlocal stochastic convolution is established. Furthermore the convergence and simulation is provided by the Galerkin finite element method as well. </abstract>

Global Existence for an Attraction–Repulsion Chemotaxis-Fluid System in a Framework of Besov–Morrey type

We consider an attraction-repulsion chemotaxis model in the whole space $${\mathbb {R}}^3$$ with logistic source coupled with the incompressible Navier–Stokes equations. By means of a contraction argument, we obtain the existence and uniqueness of global mild solutions in a framework of Besov type, namely Besov spaces based on Morrey spaces. In comparison with previous results for the system dealt with, that framework provides new solutions and allows us to consider larger initial data and force classes for global existence and uniqueness. To carry out our results, we prove some essential lemmas and estimates related to the heat semigroup and continuity properties for the Bony decomposition in our setting.

Global Mild Solutions for a Nonautonomous 2D Navier–Stokes Equations with Impulses at Variable Times

The present paper deals with existence and uniqueness of global mild solutions for the 2D Navier-Stokes equations with impulses. Using the framework of nonautonomous dynamical systems, we extend previous results considering the 2D Navier-Stokes equations with impulse effects and allowing that the nonlinear terms are explicitly time-dependent. Additionally, we present sufficient conditions to obtain dissipativity (boundedness) for solutions starting in bounded sets.

Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction–diffusion equations with delay

In this paper, we study the mild solutions of a class of nonlinear fractional reaction–diffusion equations with delay and Caputo’s fractional derivatives. By the measure of noncompactness, the theory of resolvent operators, the fixed point theorem and the Banach contraction mapping principle, we develop various theorems for the existence and uniqueness of the global mild solutions for the problem.

Stochastic nonlinear wave equation with memory driven by compensated Poisson random measures

In this paper, we study a class of stochastic nonlinear wave equation with memory driven by Lévy noise. We first show the existence and uniqueness of global mild solutions using a suitable energy function. Second, under some additional assumptions we prove the exponential stability of the solutions.

STOCHASTIC CAHN–HILLIARD EQUATION WITH FRACTIONAL NOISE

We study the existence and uniqueness of global mild solutions to a class of stochastic Cahn–Hilliard equations driven by fractional noises (fractional in time and white in space), through a weak convergence argument.

State Trajectories Analysis for a Class of Tubular Reactor Nonlinear Nonautonomous Models

The existence and uniqueness of global mild solutions are proven for a class of semilinear nonautonomous evolution equations. Moreover, it is shown that the system, under considerations, has a unique steady state. This analysis uses, essentially, the dissipativity, a subtangential condition, and the positivity of the relatedC0-semigroup.

Global existence, uniqueness, and continuous dependence for a semilinear initial value problem

In this paper, we consider the semilinear initial value problem associated with an operator A whose spectrum lies in a sector of the complex plane and whose resolvent satisfies ‖(z−A)−1‖⩽M|z|γ for some −1<γ<0 and all z outside the sector. The properties of existence and uniqueness of global mild solutions and continuous dependence on the initial data are investigated.