DCDS
Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source
Xie Li Zhaoyin Xiang
Discrete & Continuous Dynamical Systems - A 2015, 35(8): 3503-3531 doi: 10.3934/dcds.2015.35.3503
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).
keywords: parabolic-parabolic Keller-Segel systems. Global existence parabolic-elliptic Keller-Segel system boundedness
DCDS-B
Boundedness in a two-species chemotaxis parabolic system with two chemicals
Xie Li Yilong Wang
Discrete & Continuous Dynamical Systems - B 2017, 22(7): 2717-2729 doi: 10.3934/dcdsb.2017132
This paper is devoted to the chemotaxis system
$\left\{\begin{aligned}&u_t=Δ u-χ\nabla·(u\nabla v), &x∈Ω,\,t>0,\\& τ v_t=Δ v-v+w, &x∈Ω,\,t>0,\\&w_t=Δ w-ξ\nabla·(w\nabla z), &x∈Ω,\,t>0,\\& τ z_t=Δ z-z+u, &x∈Ω,\,t>0,\end{aligned}\right.$
which models the interaction between two species in presence of two chemicals, where
$χ, \, ξ∈\mathbb{R}$
,
$Ω\subset\mathbb{R}^2$
are bounded domains with smooth boundary. It is shown that under the homogeneous Neumann boundary conditions the system possesses a unique global classical solution which is bounded whenever both
$\int_Ω u_0dx$
and
$\int_Ω w_0dx$
are appropriately small. In particular, we extend the recent results obtained by Tao and Winkler (2015, Disc. Cont. Dyn. Syst. B) to the fully parabolic case, i.e., the case of
$τ=1$
.
keywords: Chemotaxis attraction repulsion blow-up
CPAA
Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation
Xie Li Zhaoyin Xiang
Communications on Pure & Applied Analysis 2014, 13(4): 1465-1480 doi: 10.3934/cpaa.2014.13.1465
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

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