This paper is concerned with the parabolic-elliptic chemotaxis system with weakly singular sensitivity and logistic source:~$ u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v^\alpha}\nabla v) +ru-\mu u^2$, $0=\Delta v-v+u,$ under the homogeneous Neumann boundary in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$ for $n\ge 2$. where $\alpha\in(0,1)$ and $\chi,r,\mu>0$. If $\alpha\in(0,\frac{n+2}{2n})$ and $\mu>\mu_0$ with $\mu_0>0$ suitably large, we give the explicit expression of the upper bound for $u$ with respect to the coefficient $\mu$ after some time, without establishing the uniformly positive bound for $v$ from below. Furthermore, by dealing with the corresponding non-singular chemotaxis system via the transformation $z=v^{1-\alpha}$, it is proved that the solution $(u,v)$ converges to $(\frac{r}{\mu},\frac{r}{\mu})$ in $L^\infty$-norm as $t\rightarrow\infty$ if $\alpha\in(0,\frac{1}{2})$ and $\mu>\mu_\star$ sufficiently large, which is moreover enjoying exponential convergence when $\alpha\in(0,\frac{n+2}{n^2+4})$.
This paper is concerned with the parabolic-elliptic chemotaxis system involving weakly singular sensitivity and a nonlocal source: $u_t=\Delta u-\chi\nabla\cdot\left(\frac{u}{v^k}\nabla v\right) +u^\alpha\left(r-\mu\int_\Omega u^\beta\,dx\right)$ and $0=\Delta v-v+u^\gamma$ under homogeneous Neumann boundary conditions...
This third paper in a series on a parabolic-elliptic chemotaxis model studies bifurcation of positive constant equilibria in one space dimension. We consider $$ \left\{ \begin{array}{ll} u_t=\Delta u-\chi_0\nabla\cdot\left(\frac{u^m}{(1+v)^\beta}\nabla v\right)+au-bu^{1+\alpha},&x\in\Omega,\\ 0=\Delta v-\mu v+\nu u^\ga...
We study the Neumann problem $u_t-\varepsilon\Delta u=e^u-1-au\ (a>1)$ on a smooth bounded domain $\Omega\subset\mathbb{R}^2$. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium $\xi_a$ is unstable, and solutions starting above $\xi_a$ blow up in finite time. Although finite-time blow-up per...
We consider the following chemotaxis-growth system, featuring density-suppressed motility and a $\theta$-logistic growth term that incorporates a strong Allee effect: \begin{equation*} \left\{ \begin{aligned} u_t&= \Delta(\gamma(v)u) + \mu u(1-u^\theta)(u-A),&x \in \Omega, \quad t>0, v_t&= \Delta v - v+ w^\beta,&x \in...
O. Tripathi, Sourav Kumar Sasmal, Manil T. Mohan· 0 citations
This paper investigates the following May--Nowak type model for viral infection in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 2$: $\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) + \kappa - u - uw - \mu \dfrac{u^{1+\alpha}}{\ln^k(u+e)}, \\[1mm] \gamma v_t = \Delta v - v + uw, \\[1mm] w_t = \De...
This paper is concerned with a no-flux initial-boundary value problem for the degenerate volume-filling chemotaxis system with source terms, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v) + f(x, u, v), \quad v_t = \Delta v + g(u,v), \quad x\in \Omega, \ t>0 \end{align*} in a smoothly bounded doma...
Osuke Shibata· 0 citations
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