## Journals

- Advances in Mathematics of Communications
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### Open Access Journals

DCDS

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.

DCDS

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.

DCDS

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.

CPAA

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.

CPAA

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.

CPAA

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.

DCDS-B

In this paper we study some mathematical models describing evolution of population density and
spread of epidemics in population systems in which spatial movement of individuals depends only
on the departure and arrival locations and does not have apparent connection with the population
density. We call such models as population migration models and migration epidemics models,
respectively. We first apply the theories of positive operators and positive semigroups to make
systematic investigation to asymptotic behavior of solutions of the population migration models
as time goes to infinity, and next use such results to study asymptotic behavior of solutions of
the migration epidemics models as time goes to infinity. Some interesting properties of solutions
of these models are obtained.

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