Global Existence Of Mild Solutions Research Articles

Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation

Abstract In this paper, we present the existence of global mild solutions for the Hilfer fractional neutral evolution equations (HFNEEs), regardless of whether the semigroups are compact or noncompact. We achieve our main results by utilizing the generalization Ascoli-Arzelà theorem, Krasnoselskii’s fixed point theorem, Laplace transform, and measures of noncompactness. To demonstrate the feasibility of our method, we provide an example.

Global solutions and their limit behavior for parabolic inclusions with an unbounded right-hand part

We study global resolvability for parabolic inclusions with an upper semicontinuous multi-valued right-hand part of more than linear growth. Theorems about the existence of global mild solutions in different phase spaces are proved. Limit sets for the obtained global solutions in the corresponding phase spaces are investigated.

On the global existence of mild solutions to binary-ternary Boltzmann equation in an integrable space

On the global existence of mild solutions to binary-ternary Boltzmann equation in an integrable space

Stochastic Quasi-Geostrophic Equation with Jump Noise in Lp Spaces

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild Lp(D)-solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers α∈(12,1] by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in L∞(0,T;Lp(D)) space for all p>22α−1, which is consistent with the deterministic case.

On the fractional chemotaxis Navier-Stokes system in the critical spaces

<p style='text-indent:20px;'>We consider the fractional chemotaxis Navier-Stokes equations which are the fractional Keller-Segel model coupled with the Navier-Stokes fluid in the whole space, and prove the existence of global mild solutions with the small critical initial data in Besov-Morrey spaces. Our results enable us to obtain the self-similar solutions provided the initial data are homogeneous functions with small norms and considering the case of chemical attractant without degradation rate. Moreover, we show the asymptotic stability of solutions as the time goes to infinity and obtain a class of asymptotically self-similar ones.</p>

Existence and stability results for a class of nonlocal semilinear equations involving delays

We investigate the existence and weak stability of mild solutions for a class of nonlocal semilinear equations involving finite delays. Based on local estimates and fixed point arguments, we prove the global existence of mild solutions to problems in whic

Continuous dependence on parameters of differential inclusion using new techniques of fixed point theory

In this paper, we establish the global existence and the continuous dependence on parameters for a set solutions to a class of time-fractional partial differential equation in the form {?/?t u(t) + KA?1 ?/?t u(t) +A?2u(t) ? F(t, u(t), ?), t ? I, u(T) = h, (resp. u(0) = h) on ?, where ?1, ?2 > 0 and I = [0, T) (resp. I = (0, T]). Precisely, first results are about the global existence of mild solutions and the compactness of the mild solutions set. These result are mainly based on some necessary estimates derived by considering the solution representation in Hilbert spaces. The remaining result is the continuous dependence of the solutions set on some special parameters. The main technique used in this study include the fixed point theory and some certain conditions of multivalued operators.

Hilfer fractional stochastic evolution equations on infinite interval

Abstract This paper concerns the global existence of mild solutions for a class of Hilfer fractional stochastic evolution equations on infinite interval (0, +∞), while the existing work were considered on finite interval. The main difficulties here are how to construct suitable Banach spaces, proper operator relations, and then how to formulate the new criteria to guarantee the global existence of mild solutions on the previous constructed spaces under non-Lipschitz conditions. We mainly rely on the generalized Ascoli–Arzela theorem we established, Wright function, Schauder’s fixed point principle, and Kuratowski’s measure of noncompactness to handle with the infinite interval problems. Moreover, we give two examples to demonstrate the feasibility and utility of our results.

Global Well-Posedness and Analyticity for the Three-Dimensional Incompressible Nematic Liquid Crystal Flows in Scaling Invariant Spaces

The Cauchy problem for the three-dimensional incompressible flows of liquid crystals in scaling invariant spaces is considered. In this work, we exhibit three results. First, we prove the global well-posedness of mild solution for the system without the supercritical nonlinearity∇d2dwhen the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients. Our second result is a proof of the global existence of mild solution in the time dependent spaces for the system including the term∇d2dfor small initial data. Lastly, we also get analyticity of the solution.

Local and global existence of mild solutions of time-fractional Navier–Stokes system posed on the Heisenberg group

This paper is a development of the results and techniques of the two papers (Carvalho-Neto and Planas in J Differ Equ 259:2948–2980, 2015; Oka in J Math Anal Appl 473:382–407, 2019) for the aim of addressing the existence and uniqueness of local and global mild solutions, on the Heisenberg group $$\mathbb {H}^d$$ , of the time-fractional Navier–Stokes system with time derivative of order $$\alpha \in (0, 1)$$ . The proof relies on Schaefer’s fixed point theorem.

Blowup and global existence of mild solutions for fractional extended Fisher–Kolmogorov equations

Abstract In this paper, we investigate the blowup, as well as global existence, and uniqueness of mild solutions for the initial-boundary value problem to a class of fractional extended Fisher–Kolmogorov equations with a general nonlinear term. We establish a general framework to find the global mild solutions for fractional extended Fisher–Kolmogorov equations with general nonlinear function, which will provide an effective way to deal with such problems. The results obtained in this paper can be considered as a contribution to this nascent field.

Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces

<p style='text-indent:20px;'>The paper is concerned with the Navier-Stokes-Nernst-Planck-Poisson system arising from electrohydrodynamics in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula>. By means of the implicit function theorem, we prove the global existence of mild solutions for Cauchy problem of this system with small initial data in critical Besov-Morrey spaces. In comparison to the previous works, our existence result provides a new class of initial data, for which the problem is global solvability. Meanwhile, based on the so-called Gevrey estimates, we verify that the obtained mild solutions are analytic in the spatial variables. As a byproduct, we show the asymptotic stability of solutions as the time goes to infinity. Furthermore, decay estimates of higher-order derivatives of solutions are deduced in Morrey spaces.

Mild solutions of the stochastic MHD equations driven by fractional Brownian motions

The incompressible magnetohydrodynamic equations driven by additive fractional Brownian motions are considered. We firstly establish the local existence and uniqueness of the mild solution in Lp space on a smooth bounded domain in Rd(d=2,3). The proof is based on the semigroup theory, fixed point theorem and the results of stochastic PDEs of linear parabolic type. In the proof, the eigenvalue problem with perfectly conducting wall condition is considered to weaken the requirements of noise terms. Finally, the global existence of mild solutions is also established by energy estimate.

Existence and uniqueness of mild solutions to the chemotaxis-fluid system modeling coral fertilization

In this paper, we consider the egg-sperm chemotaxis model of coral with the incompressible fluid equations in the whole space. The existence of global mild solutions in scaling invariant spaces is proved with sufficient small initial data. Here the main tool we use is the implicit function theorem. Furthermore, we obtain the asymptotic stability of solutions when the time goes to infinity. Since the initial data could be in the weak L^{p}-spaces, we finally get the existence of self-similar solutions when the initial data are small homogeneous functions.

The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise

This article is devoted to study time fractional stochastic evolution inclusions with infinite delays driven by a nonlinear multiplicative noise and a fractional Brownian motion with Hurst parameter \begin{document}$ H\in(\frac{1}{2},1) $\end{document} . First of all, we investigate the local and global existence of mild solutions to such evolution inclusions by using the fractional resolvent operator theory and some new results on the measure of noncompactness for the stochastic integral term. Further, we prove that every mild solution decays exponentially to zero in the sense of mean-square topology.

The Attractors of Camassa–Holm Equation in Unbounded Domains

In this paper, we deal with the existence of global mild solutions and asymptotic behavior to the viscous Camassa–Holm equation in the locally uniform spaces. First we establish the global well-posedness for the Cauchy problem of viscous Camassa–Holm equation in $${\mathbb {R}}^1$$ for any initial data $$u_0\in {\dot{H}}^1_U({\mathbb {R}}^1).$$ Then we study the long time dynamical behavior of non-autonomous viscous Camassa–Holm equation on $${\mathbb {R}}^1$$ with a new class of external forces and show the existence of $$(H^1_U({\mathbb {R}}^1),H^1_\phi ({\mathbb {R}}^1))$$ -uniform(w.r.t. $$g\in \mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)$$ ) attractor $$\mathcal {A}_{\mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)}$$ with locally uniform external forces being translation uniform bounded but not translation compact in $$L_b^2({\mathbb {R}};L^2_U({\mathbb {R}}^1))$$ .

Existence of global mild solutions for a class of fractional partial functional differential equations

This paper is concerned with a class of fractional partial functional differential equations with nonlocal conditions in Banach spaces. By using the theories of resolvent operator, the Schauder fixed point theorem and the Banach contraction mapping principle, we obtain the sufficient conditions for the local and global existence of the mild solution.

Global Solutions to 3D Rotating Boussinesq Equations in Besov Spaces

In this paper, the three-dimensional rotating Boussinesq equations are considered. By striking new balances between the regularizing effects of the Laplacian dissipation and the dispersive effects of the Coriolis force, we obtain the global existence of mild solutions to Cauchy problem of the three-dimensional rotating Boussinesq equations. Particularly, the size of the norm of initial velocity can be arbitrarily large, provided that the speed of rotation is sufficiently high.

Existence and uniqueness of mild solutions to the incompressible nematic liquid crystal flow

In this paper, we consider the incompressible nematic liquid crystal flow in the whole space, and obtain the existence of global mild solutions with small initial data in the scaling invariant space. The main tool we have used is the implicit function theorem. Furthermore, we establish the asymptotic stability of solutions when the time goes to infinity. And in view of our construction of solutions in the weak Lp-spaces, we finally derive the existence of self-similar solutions provided the initial data are small homogeneous functions.

Global existence results for second order functional differential equations with delay

Our aim in this work is to study the existence of global mild solutions of a second order functional differential equation with delay. We use the Schauder fixed point theorem combined with the cosine family of linear bounded operators for the existence of solutions.