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### Open Access Journals

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

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.

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