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
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
$χ, \, ξ∈\mathbb{R}$
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
$\int_Ω u_0dx$
$\int_Ω w_0dx$
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
keywords: Chemotaxis attraction repulsion blow-up
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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