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DCDS-B

In this paper, we consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-parabolic type
\begin{equation*}
\left\{
\begin{split}
&u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\qquad & x\in\Omega,\,\, t>0,\\
&v_t=\Delta v+\alpha u-\beta v,\qquad &x\in\Omega, \,\,t>0,\\
&w_t=\Delta w+\gamma u-\delta w,\qquad &x\in\Omega,\,\, t>0
\end{split}
\right.
\end{equation*}
under homogeneous Neumann boundary conditions, where $D(u)\geq c_D (u+\varepsilon)^{m-1}$ and $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary. It is shown that whenever $m>1$, for any sufficiently smooth nonnegative initial data, the system admits a global bounded classical solution for the case of non-degenerate diffusion (i.e., $\varepsilon>0$), while the system possesses a global bounded weak solution for the case of degenerate diffusion (i.e., $\varepsilon=0$).

CPAA

This paper deals with the blow-up properties of solutions to a
degenerate parabolic system coupled via nonlinear boundary flux.
Firstly, we construct the self-similar supersolution and
subsolution to obtain the critical global existence curve.
Secondly, we establish the precise blow-up rate estimates for solutions which blow up
in a finite time. Finally, we investigate the localization of blow-up points. The critical
curve of Fujita type is conjectured with the aid of some new results.

CPAA

In this paper, we study the existence of local/global solutions to the Cauchy problem
\begin{eqnarray}
\rho(x)u_t=\Delta u+q(x)u^p, (x,t)\in R^N \times (0,T),\\
u(x,0)=u_{0}(x)\ge 0, x \in R^N
\end{eqnarray}
with $p > 0$ and $N\ge 3$. We describe the sharp decay conditions on $\rho, q$ and the data $u_0$ at infinity that guarantee the local/global existence of nonnegative solutions.

keywords:
global existence
,
Fujita exponent
,
Cauchy problem
,
inhomogeneous heat equation.
,
Local existence

DCDS

In this paper, we investigate the quasilinear Keller-Segel equations (q-K-S):
\[
\left\{
\begin{split}
&n_t=\nabla\cdot\big(D(n)\nabla n\big)-\nabla\cdot\big(\chi(n)\nabla c\big)+\mathcal{R}(n), \qquad x\in\Omega,\,t>0,\\
&\varrho c_t=\Delta c-c+n, \qquad x\in\Omega,\,t>0,
\end{split}
\right.
\]
under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$. For both $\varrho=0$ (parabolic-elliptic case) and $\varrho>0$ (parabolic-parabolic case), we will show the global-in-time existence and uniform-in-time boundedness of solutions to equations (q-K-S) with both non-degenerate and

*degenerate*diffusions on the*non-convex*domain $\Omega$, which provide a supplement to the dichotomy boundedness vs. blow-up in parabolic-elliptic/parabolic-parabolic chemotaxis equations with degenerate diffusion, nonlinear sensitivity and logistic source. In particular, we improve the recent results obtained by Wang-Li-Mu (2014, Disc. Cont. Dyn. Syst.) and Wang-Mu-Zheng (2014, J. Differential Equations).
CPAA

In this paper, we consider a reaction-diffusion system coupled by
nonlinear memory. Under appropriate hypotheses, we prove that the
solution either exists globally or blows up in finite time.
Furthermore, the blow-up rate estimates are obtained.

CPAA

This paper deals with the blow-up properties and asymptotic
behavior of solutions to a semilinear integrodifferential system
with nonlocal reaction terms in space and time. The blow-up
conditions are given by a variant of the eigenfunction method
combined with new properties on systems of differential
inequalities. At the same time, the blow-up set is obtained. For
some special cases, the asymptotic behavior of the blow-up
solution is precisely characterized.

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