Existence Of Local Mild Solutions Research Articles

On a Class of Stochastic Damped Wave Equation

The present work considers a wave equation with multiplicative Gaussian white noise and weak dissipative term on a bounded domain. We first give a theorem including the local existence of mild solutions. An energy bound and a differential inequality are used to give sufficient conditions that provide the blow-up of mild local solutions of the stochastic wave equation. The paper's main contribution comes from handling a multiplicative noise and a general source term contrary to the articles that exist in the literature.

Suppression of epitaxial thin film growth by mixing

We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the case 2<p<3:∂tu+(−Δ)2u=−∇⋅(|∇u|p−2∇u). The study of this form of equations arises from mathematical models that simulate the epitaxial growth of the thin film. We prove the local existence of mild solutions for any initial data lies in L2 in both cases. Our main result is: in the advective case, if the imposed advection is sufficiently mixing, then the global existence of solution can be proved, and the solution will converge exponentially to a homogeneous mixed state. While in the absence of advection, there exist initial data in H2∩W1,∞ such that the solution will blow up in finite time.

Large time behaviour for the motion of a solid in a viscous incompressible fluid

In this article, we study the long-time behaviour of a system describing the coupled motion of a rigid body and of a viscous incompressible fluid in which the rigid body is contained. We assume that the system formed by the rigid body and the fluid fills the entire space $${\mathbb {R}}^3$$ . In the case in which the rigid body is a ball, we prove the local existence of mild solutions and, when the initial data are small, the global existence of solutions for this system with a precise description of their large time behavior. Our main result asserts, in particular, that if the initial datum is small enough in suitable norms then the position of the center of the rigid ball converges to some $$h_\infty \in {\mathbb {R}}^3$$ as time goes to infinity. This result contrasts with those known for the analogues of our system in 2 or 1 space dimensions, where it has been proved that the body quits any bounded set, provided that we wait long enough. To achieve this result, we use a “monolithic” type approach, which means that we consider a linearized problem in which the equations of the solid and of the fluid are still coupled. An essential role is played by the properties of the semigroup, called fluid-structure semigroup, associated to this coupled linearized problem. The generator of this semigroup is called the fluid-structure operator. Our main tools are new $$L^p - L^q$$ estimates for the fluid-structure semigroup. Note that these estimates are proved for bodies of arbitrary shape. The main ingredients used to study the fluid-structure semigroup and its generator are resolvent estimates which provide both the analyticity of the fluid-structure semigroup (in the spirit of a classical work of Borchers and Sohr) and $$L^p- L^q$$ decay estimates (by adapting a strategy due to Iwashita).

Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection

We study the Kuramoto-Sivashinsky equation (KSE) in scalar form on the two-dimensional torus with and without advection by an incompressible vector field. We prove local existence of mild solutions for arbitrary data in L 2. We then study the issue of global existence. We prove global existence for the KSE in the presence of advection for arbitrary data, provided the advecting velocity field v satisfies certain conditions that ensure the dissipation time of the associated hyperdiffusion-advection equation is sufficiently small. In the absence of advection, global existence can be shown only if the linearized operator does not admit any growing mode and for sufficiently small initial data.

Local and blowing-up solutions for an integro-differential diffusion equation and system

In the present paper initial problems for the semilinear integro-differential diffusion equation and system are considered. The analogue of Duhamel principle for the linear integro-differential diffusion equation is proved. The results on existence of local mild solutions and Fujita-type critical exponents to the semilinear integro-differential diffusion equation and system are presented.

Initial value problem for fractional Volterra integro-differential equations with Caputo derivative

&lt;p style='text-indent:20px;'&gt;In this paper, we consider the time-fractional Volterra integro-differential equations with Caputo derivative. For globally Lispchitz source term, we investigate the global existence for a mild solution. The main tool is to apply the Banach fixed point theorem on some new weighted spaces combining some techniques on the Wright functions. For the locally Lipschitz case, we study the existence of local mild solutions to the problem and provide a blow-up alternative for mild solutions. We also establish the problem of continuous dependence with respect to initial data. Finally, we present some examples to illustrate the theoretical results.&lt;/p&gt;

Well-posedness of an initial value problem for fractional diffusion equation with Caputo–Fabrizio derivative

In this paper, we consider the following Cauchy problem of space–time fractional diffusion equations CFD0,tαu+(−L)su=G(t,x;u),in(0,T]×Ω,u(t,x)=0,on(0,T]×∂Ω,u(0,x)=u0(x),inΩ,The time fractional derivative is taken in the sense of Caputo-Fabrizio type. We derive representation of solutions by using Laplace transform and we further establish the existence and uniqueness of the mild solution. Both the linear case and the nonlinear case with globally Lipschitz assumption, we investigate the regularity of the mild solution. For the case of locally Lipschitz assumption, we study the existence of local mild solutions to the problem and then a blow-up alternative is established. We also consider the problem of continuous dependence with respect to initial data.

A blowup alternative result for fractional nonautonomous evolution equation of Volterra type

In this article, we consider a class of fractional non-autonomous integro-differential evolution equation of Volterra type in a Banach space $E$, where the operators in linear part (possibly unbounded) depend on time $t$. Combining the theory of fractional calculus, operator semigroups and measure of noncompactness with Sadovskii's fixed point theorem, we firstly proved the local existence of mild solutions for corresponding fractional non-autonomous integro-differential evolution equation. Based on the local existence result and a piecewise extended method, we obtained a blowup alternative result for fractional non-autonomous integro-differential evolution equation of Volterra type. Finally, as a sample of application, these results are applied to a time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition. This paper is a continuation of Heard and Rakin [13, J. Differential Equations, 1988] and the results obtained essentially improve and extend some related conclusions in this area.

On the well‐posedness of a Volterra equation with applications in the Navier‐Stokes problem

This paper is dedicated to the study of a family of nonlinear Volterra equations coming from the theory of viscoelasticity. We analyze the existence of local mild solutions to the problem and their possible continuation to a maximal interval of existence. We also consider the problem of continuous dependence with respect to initial data.

Stochastic functional differential equations of Sobolev-type with infinite delay

Stochastic differential equations have been widely used to model a number of phenomena in diverse fields of science and engineering. In this paper, we focus on the local existence of mild solution for a class of stochastic functional differential equations of Sobolev-type with infinite delay. Furthermore, the results are extended to study the local existence results for neutral stochastic differential equations of Sobolev-type. Using the semigroup theory and fixed point argument, we establish a set of sufficient conditions for obtaining the required result. Furthermore, existence results for integro-differential equations of Sobolev-type is also discussed. Finally, an example is provided to illustrate the obtained theory.

Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results

In this work we study several questions concerning to abstract fractional Cauchy problems of order $\alpha\in(0,1)$. Concretely, we analyze the existence of local mild solutions for the problem, and its possible continuation to a maximal interval of existence. The case of critical nonlinearities and corresponding regular mild solutions is also studied. Finally, by establishing some general comparison results, we apply them to conclude the global well-posedness of a fractional partial differential equation coming from heat conduction theory.

Well-posedness of the Navier—Stokes—Maxwell equations

We study the local and global well-posedness of a full system of magnetohydrodynamic equations. The system is a coupling of the incompressible Navier—Stokes equations with the Maxwell equations through the Lorentz force and Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale-invariant spaces classically used for Navier—Stokes. These solutions are global if the initial data are small enough. Our results not only simplify and unify the proofs for the space dimensions 2 and 3, but also refine those in [8]. The main simplification comes from an a prioriLt2 (Lx∞) estimate for solutions of the forced Navier—Stokes equations.

On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces

We study the Navier-Stokes system with initial data belonging to sum of twoweak-$L^{p}$ spaces, which contains the sum of homogeneous function withdifferent degrees. The domain $\Omega$ can be either an exterior domain, the half-space, the whole space or a bounded domain with dimension $n\geq 2$. We obtain the existence of local mild solutions in the sameclass of initial data and moreover we show results about uniqueness,regularity and continuous dependence of solutions with respect to the initial data. To obtain our results we prove a new Hölder-type inequality on the sum of Lorentz spaces.

Existence Results for Second Order Differential Equations with Nonlocal Conditions in Banach Spaces

This work is concerned with implicit second order abstract differential equations with nonlocal conditions. Assuming that the involved operators satisfy some compactness properties, we establish the existence of local mild solutions, the existence of global mild solutions and the existence of asymptotically almost periodic solutions.