DCDS
Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord
Fujun Zhou Shangbin Cui
Discrete & Continuous Dynamical Systems - A 2008, 21(3): 929-943 doi: 10.3934/dcds.2008.21.929
In this paper we study a multidimensional moving boundary problem modeling the growth of tumor cord. This problem contains two coupled elliptic equations defined in a bounded domain in $R^2$ whose boundary consists of two disjoint closed curves, one fixed and the other moving and a priori unknown. The evolution of the moving boundary is governed by a Stefan type equation. By using the functional analysis method based on applications of the theory of analytic semigroups, we prove that (1) this problem is locally well-posed in Hölder spaces, (2) it has a unique radially symmetric stationary solution, and (3) this radially symmetric stationary solution is asymptotically stable for arbitrary sufficiently small perturbations in these Hölder spaces.
keywords: stability. tumor cord Moving boundary problem well-posedness
DCDS
Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations
Junde Wu Shangbin Cui
Discrete & Continuous Dynamical Systems - A 2009, 24(2): 625-651 doi: 10.3934/dcds.2009.24.625
We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value γ* > 0 for the surface tension coefficient $\gamma$, such that in the case γ > γ* this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable.
keywords: stationary solution Free boundary problem asymptotic stability. Stokes equations tumor growth
DCDS
Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth
Junde Wu Shangbin Cui
Discrete & Continuous Dynamical Systems - A 2010, 26(2): 737-765 doi: 10.3934/dcds.2010.26.737
In this paper we first study local phase diagram of an abstract parabolic differential equation in a Banach space such that the equation possesses an invariance structure under a local Lie group action. Next we use this abstract result to study a free boundary problem modeling the growth of non-necrotic tumors in the presence of inhibitors. This problem contains two reaction-diffusion equations describing the diffusion of the nutrient and the inhibitor, respectively, and an elliptic equation describing the distribution of the internal pressure. There is also an equation for the surface tension to govern the movement of the free boundary. By first performing some reduction processes to write this free boundary problem into a parabolic differential equation in a Banach space, and next using a new center manifold theorem established recently by Cui [8], and the abstract result mentioned above, we prove that under suitable conditions the radial stationary solution is locally asymptotically stable under small non-radial perturbations, and when these conditions are not satisfied then such a stationary solution is unstable. In the second case we also give a description of local phase diagram of the equation in a neighborhood of the radial stationary solution and construct its stable and unstable manifolds. In particular, we prove that in the unstable case, if the transient solution exists globally and is contained in a neighborhood of the radial stationary solution, then the transient solution will finally converge to a nearby radial stationary solution uniquely determined by the initial data.
keywords: stable/unstable manifold asymptotic stability invariance structure. tumor growth Free boundary problem
CPAA
Regularizing rate estimates for mild solutions of the incompressible Magneto-hydrodynamic system
Qiao Liu Shangbin Cui
Communications on Pure & Applied Analysis 2012, 11(5): 1643-1660 doi: 10.3934/cpaa.2012.11.1643
We establish some regularizing rate estimates for mild solutions of the magneto-hydrodynamic system (MHD). These estimations ensure that there exist positive constants $K_1$ and $K_2$ such that for any $\beta\in\mathbb{Z}^{n}_{+}$ and any $t\in (0,T^\ast)$, where $T^\ast$ is the life-span of the solution, we have $\| (\partial_{x}^{\beta}u(t),\partial_{x}^{\beta}b(t))\|_{q}\leq K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2} -\frac{n}{2}(\frac{1}{n}-\frac{1}{q})}$. Spatial analyticity of the solution and temporal decay of global solutions are direct consequences of such estimates.
keywords: Magneto-hydrodynamic system regularizing rate spatial analyticity. mild solution
CPAA
Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows
Jihong Zhao Qiao Liu Shangbin Cui
Communications on Pure & Applied Analysis 2013, 12(1): 341-357 doi: 10.3934/cpaa.2013.12.341
In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solution. In order to figure out the relation between the solution obtained here and weak solutions of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.
keywords: Liquid crystal flow stability global existence blow up weak-strong uniqueness.
CPAA
Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors
Fujun Zhou Junde Wu Shangbin Cui
Communications on Pure & Applied Analysis 2009, 8(5): 1669-1688 doi: 10.3934/cpaa.2009.8.1669
In this paper we study a moving boundary problem modeling the growth of multi-layer tumors under the action of inhibitors. The problem contains two coupled reaction-diffusion equations and one elliptic equation defined on a strip-like domain in $R^n$, with one part of the boundary moving and a priori unknown. The evolution of the moving boundary is governed by a Stefan type equation, with the surface tension effect taken into consideration. Local existence and asymptotic behavior of solutions to this problem are investigated. The analysis is based on the employment of the functional analysis method combing with the well-posedness and geometric theory for parabolic differential equations in Banach spaces.
keywords: Moving boundary problem multi-layer tumors asymptotic behavior.

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