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 quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system
Yilong Wang Zhaoyin Xiang
Discrete & Continuous Dynamical Systems - B 2016, 21(6): 1953-1973 doi: 10.3934/dcdsb.2016031
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., $\varepsilon=0$).
keywords: attraction-repulsion boundedness. Parabolic-parabolic chemotaxis global existence
Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux
Chunlai Mu Zhaoyin Xiang
Communications on Pure & Applied Analysis 2007, 6(2): 487-503 doi: 10.3934/cpaa.2007.6.487
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.
keywords: Degenerate parabolic system critical global existence curve critical Fujita curve blow-up rate estimates blow-up sets.
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
Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system
Qiong Chen Chunlai Mu Zhaoyin Xiang
Communications on Pure & Applied Analysis 2006, 5(3): 435-446 doi: 10.3934/cpaa.2006.5.435
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.
keywords: blow-up set Integrodifferential system asymptotic behavior. blow-up
Global existence and blow-up to a reaction-diffusion system with nonlinear memory
Lili Du Chunlai Mu Zhaoyin Xiang
Communications on Pure & Applied Analysis 2005, 4(4): 721-733 doi: 10.3934/cpaa.2005.4.721
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.
keywords: blow-up reaction-diffusion system nonlinear memory Global existence

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