Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.     

Local solvability of the Keller–Segel system with Cauchy data in the Besov spaces

The Cauchy problem for the Keller–Segel system of parabolic elliptic type is considered for initial data in the Besov spaces with p < ∞ , and a sufficient condition is given on the existence and the uniqueness of local solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Possibility of the existence of blow‐up solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear ‘degenerate’ Keller–Segel system of parabolic–parabolic type under the super‐critical condition. In the ‘non‐degenerate’ case, Winkler (Math. Methods Appl. Sci. 2010; 33:12–24) constructed the initial data such that the solution blows up in either finite or infinite time. However, the blow‐up under the super‐critical condition is left as an open question in the ‘degenerate’ case. In this paper, we try to give an answer to the question under assuming the existence of local solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence and asymptotic behaviour for the time‐fractional Keller–Segel model for chemotaxis

One of the most important systems for understanding chemotactic aggregation is the Keller–Segel system. We consider the time‐fractional Keller–Segel system of order . We prove an existence result with small initial data in a class of Besov–Morrey spaces. Self‐similar solutions are obtained and we also show an asymptotic behaviour result.     

Global existence and time decay estimate of solutions to the Keller–Segel system

We consider the chemotaxis‐Navier–Stokes system 1.1-1.4 (Keller–Segel system) in the whole space, which describes the motion of oxygen‐driven bacteria, eukaryotes, in a fluid. We proved the global existence and time decay estimate of solutions to the Cauchy problem 1.1-1.2 in with the small initial data. Moreover, when the fluid motion is described by the Stokes equations, we established the global weak solutions to 1.3-1.4 in with the potential function ? is small and the initial density n₀(x) has finite mass.     

Well‐posedness for fractional Navier–Stokes equations in the largest critical spaces

This note studies the well‐posedness of the fractional Navier–Stokes equations in some supercritical Besov spaces as well as in the largest critical spaces for β ∈ (1/2,1). Meanwhile, the well‐posedness for fractional magnetohydrodynamics equations in these Besov spaces is also studied. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line

In this paper, we are interested in a model derived from the 1‐D Keller‐Segel model on the half line x >  as follows: where l is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of l > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of l < 0. Copyright © 2016 John Wiley & Sons, Ltd.     

Local well‐posedness for the Cauchy problem of the MHD equations with mass diffusion

This paper studies the Cauchy problem of the MHD equations with mass diffusion. We use the Tikhonov fixed point theorem to prove a local‐in‐time well‐posedness theorem. Copyright © 2010 John Wiley & Sons, Ltd.     

Local well‐posedness and blow‐up criteria of solutions for a rod equation

In this paper we consider a new rod equation derived recently by Dai [Acta Mech. 127 No. 1–4, 193–207 (1998)] for a compressible hyperelastic material. We establish local well‐posedness for regular initial data and explore various sufficient conditions of the initial data which guarantee the blow‐up in finite time both for periodic and non‐periodic case. Moreover, the blow‐up time and blow‐up rate are given explicitly. Some interesting examples are given also. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global well‐posedness of the 2D Euler‐Boussinesq system with stratification effects

This paper is concerned with the Cauchy problem of the two‐dimensional Euler–Boussinesq system with stratification effects. We obtain the global existence of a unique solution to this system without assumptions of small initial data in Sobolev spaces. Copyright © 2017 John Wiley & Sons, Ltd.     

On well‐posedness for some thermo‐piezoelectric coupling models

There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost‐effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor, the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well‐posedness of such models. There is a need to extend the use of such sensors into high‐temperature environments, and so this paper shows, for a broad class of models, the well‐posedness of the magneto‐electro‐thermo‐elastic problem. Because of its widespread use in the literature, we also show the well‐posedness of the quasi‐electrostatic case. Copyright © 2016 John Wiley & Sons, Ltd.     

On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces

This paper is devoted to the study of the Cauchy problem of incompressible magneto‐hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n?3, we establish the global well‐posedness of the Cauchy problem of an incompressible magneto‐hydrodynamics system for small data and the local one for large data in the Besov space ? (?ⁿ), 1?p<∞ and 1?r?∞. Meanwhile, we also prove the weak–strong uniqueness of solutions with data in ? (?ⁿ)∩L²(?ⁿ) for n/2p+2/r>1. In the case of n=2, we establish the global well‐posedness of solutions for large initial data in homogeneous Besov space ? (?²) for 2<p<∞ and 1?r<∞. Copyright © 2008 John Wiley & Sons, Ltd.     

On the global well‐posedness for the two‐dimensional Boussinesq equations with horizontal dissipation

In this paper, we study the global well‐posedness for the two‐dimensional nonlinear Boussinesq equations with horizontal dissipation. The method we adopted is the smoothing effect in horizontal direction and the low‐high decomposition technique.     

Strong well‐posedness of a three phase problem with nonlinear transmission condition

We prove existence and uniqueness of strong solutions to a quasilinear parabolic‐elliptic system modelling an ionic exchanger. This chemical system consists of three phases connected with nonlinear boundary conditions. The most interesting difficulty of our problem manifests in the nonlinear transmission condition, as almost all quantities are non‐linearly involved in this boundary equation. Our approach is based on the contraction mapping principle, where maximal L_p‐regularity of the associated linear problem is used to obtain a fixed point equation of the starting problem. Copyright © 2012 John Wiley & Sons, Ltd.     

Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions

In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space L^p( R ⁿ). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in H^s(s≥0) and ill posed in H^s(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in L^p( R ²) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit p_c=1. In one‐dimensional case, we show that the problem is locally well posed in L¹( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.     

On the well‐posedness of a mathematical model of quorum‐sensing in patchy biofilm communities

We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Global well‐posedness of the Cauchy problem for certain magnetohydrodynamic‐α models

This paper is devoted to study the Cauchy problem for certain incompressible magnetohydrodynamics‐α model. In the Sobolev space with fractional index s>1, we proved the local solutions for any initial data, and global solutions for small initial data. Furthermore, we also prove that as α→0, the MHD‐α model reduces to the MHD equations, and the solutions of the MHD‐α model converge to a pair of solutions for the MHD equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Local well‐posedness and ill‐posedness for the fractal Burgers equation in homogeneous Sobolev spaces

In this paper, we consider local well‐posedness and ill‐posedness questions for the fractal Burgers equation. First, we obtain the well‐posedness result in the critical Sobolev space. We also present an unconditional uniqueness result. Second, we show the ill‐posedness from the point of view of the Picard iterations in the supercritical space. Copyright © 2008 John Wiley & Sons, Ltd.     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.