374 research outputs found
A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity
We consider the parabolic chemotaxis model in a
smooth, bounded, convex two-dimensional domain and show global existence and
boundedness of solutions for for some , thereby
proving that the value is not critical in this regard. Our main tool
is consideration of the energy functional for ,
, where using nonzero values of appears to be new in this context.Comment: 11 page
Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity
We prove convergence of positive solutions to in a bounded domain , ,
with smooth boundary in the case of and identify the
-limit of as as the solution of the
corresponding stationary problem. This behaviour is different from the cases of
which are known to result in
convergence to zero or blow-up in finite time, respectively.
The proof is based on a monotonicity property of
along trajectories and the analysis of an associated constrained minimization
problem.
Keywords: degenerate diffusion, nonlocal nonlinearity, long-term behaviou
Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system
We consider a parabolic-elliptic chemotaxis system generalizing
in bounded smooth domains , , and with homogeneous Neumann boundary conditions. We
show that
*) solutions are global and bounded if
*) solutions are global if
*) close to given radially symmetric functions there are many initial data
producing unbounded solutions if .
In particular, if and , there are
many initial data evolving into solutions that blow up after infinite time
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion
This article deals with an initial-boundary value problem for the coupled
chemotaxis-haptotaxis system with nonlinear diffusion \begin{align*}
u_t=&\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla
v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w),\\ v_t=&\Delta v-v+u,\\
w_t=&-vw\end{align*} under homogeneous Neumann boundary conditions in a bounded
smooth domain , , where and
are given nonnegative parameters. The diffusivity is assumed to
satisfy for all with some . It is
proved that for sufficiently regular initial data global bounded solutions
exist whenever . For the case of non-degenerate diffusion
(i.e. ) the solutions are classical; for the case of possibly
degenerate diffusion (), the existence of bounded weak solutions is
shown
Continuation beyond interior gradient blow-up in a semilinear parabolic equation
It is known that there is a class of semilinear parabolic equations for which
interior gradient blow-up (in finite time) occurs for some solutions. We
construct a continuation of such solutions after gradient blow-up. This
continuation is global in time and we give an example when it never becomes a
classical solution again
Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions
Previous studies of chemotaxis models with consumption of the chemoattractant
(with or without fluid) have not been successful in explaining pattern
formation even in the simplest form of concentration near the boundary, which
had been experimentally observed.
Following the suggestions that the main reason for that is usage of
inappropriate boundary conditions, in this article we study solutions to the
stationary chemotaxis system in bounded domains , ,
under no-flux boundary conditions for and the physically meaningful
condition on , with given parameter and
satisfying , on . We prove existence and uniqueness of solutions for any given mass
. These solutions are non-constant
Boundedness of solutions to a virus infection model with saturated chemotaxis
We show global existence and boundedness of classical solutions to a virus
infection model with chemotaxis in bounded smooth domains of arbitrary
dimension and for any sufficiently regular nonnegative initial data and
homogeneous Neumann boundary conditions.
More precisely, the system considered is
with , and solvability and boundedness
of the solution are shown under the condition that \[ \begin{cases} \alpha >
\frac 12 + \frac{n^2}{6n+4}, &\text{if } \quad 1 \leq n \leq 4 \\ \alpha >
\frac {n}4, &\text{if } \quad n \geq 5. \end{cases} \
Lack of smoothing for bounded solutions of a semilinear parabolic equation
We study a semilinear parabolic equation that possesses global bounded weak
solutions whose gradient has a singularity in the interior of the domain for
all . The singularity of these solutions is of the same type as the
singularity of a stationary solution to which they converge as
Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption
Assuming that , and
, we prove global existence of classical solutions to a
chemotaxis system slightly generalizing in a bounded domain , with
homogeneous Neumann boundary conditions and for widely arbitrary positive
initial data. In the spatially one-dimensional setting, we prove global
existence and, moreover, boundedness of the solution for any , ,
.Comment: 25 page
On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms
Introducing a suitable solution concept, we show that in bounded smooth
domains , , the initial boundary value
problem for the chemotaxis system \begin{align*}
u_t&=\Delta u -\chi\nabla\cdot\left(\frac{u}{v}\nabla v\right)+\kappa u -\mu
u^2,\\ v_t&=\Delta v -uv, \end{align*} with homogeneous Neumann boundary
conditions and widely arbitrary initial data has a generalized global solution
for any
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