Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity

We prove convergence of positive solutions to $$u_t = u\Delta u + u\int_{\Omega} |\nabla u|^2, \qquad u\rvert_{\partial\Omega} =0, \qquad u(\cdot,0)=u_0$$ in a bounded domain $\Omega\subset \mathbb{R}^n$, $n\ge 1$, with smooth boundary in the case of $\int_\Omega u_0=1$ and identify the $W_0^{1,2}(\Omega)$-limit of $u(t)$ as $t\to \infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\int_\Omega u_01$ which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of $\int_{\Omega} |\nabla u|^2$ along trajectories and the analysis of an associated constrained minimization problem. Keywords: degenerate diffusion, nonlocal nonlinearity, long-term behaviou

A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity

We consider the parabolic chemotaxis model $$u_t=\Delta u - \chi \nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=\Delta v - v + u$$ in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for $\chi\in(0,\chi_0)$ for some $\chi_0>1$, thereby proving that the value $\chi=1$ is not critical in this regard. Our main tool is consideration of the energy functional $$\mathcal{F}_{a,b}(u,v)=\int_\Omega u\ln u - a \int_\Omega u\ln v + b \int_\Omega |\nabla \sqrt{v}|^2$$ for $a>0$, $b\geq 0$, where using nonzero values of $b$ appears to be new in this context.Comment: 11 page

Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system

We consider a parabolic-elliptic chemotaxis system generalizing $$\begin{cases}\begin{split} & u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u(u+1)^{\sigma-1}\nabla v)\\ & 0 = \Delta v - v + u \end{split}\end{cases}$$ in bounded smooth domains $\Omega\subset \mathbb{R}^N$, $N\ge 3$, and with homogeneous Neumann boundary conditions. We show that *) solutions are global and bounded if ${\sigma}<m-\frac{N-2}N$ *) solutions are global if $\sigma \le 0$ *) close to given radially symmetric functions there are many initial data producing unbounded solutions if $\sigma >m-\frac{N-2}N$. In particular, if ${\sigma}\le 0$ and $\sigma > m-\frac{N-2}N$, there are many initial data evolving into solutions that blow up after infinite time

Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion \begin{align*} u_t=&\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w),\\ v_t=&\Delta v-v+u,\\ w_t=&-vw\end{align*} under homogeneous Neumann boundary conditions in a bounded smooth domain $\Omega\subset\mathbb{R}^n$, $n=2, 3, 4$, where $\chi, \xi$ and $\mu$ are given nonnegative parameters. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geq\delta u^{m-1}$ for all $u>0$ with some $\delta>0$. It is proved that for sufficiently regular initial data global bounded solutions exist whenever $m>2-\frac{2}{n}$. For the case of non-degenerate diffusion (i.e. $D(0)>0$) the solutions are classical; for the case of possibly degenerate diffusion ($D(0)\geq 0$), the existence of bounded weak solutions is shown

Continuation beyond interior gradient blow-up in a semilinear parabolic equation

It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is global in time and we give an example when it never becomes a classical solution again

Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions

Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed. Following the suggestions that the main reason for that is usage of inappropriate boundary conditions, in this article we study solutions to the stationary chemotaxis system $$\begin{cases} 0 = \Delta n - \nabla\cdot(n\nabla c) \\ 0 = \Delta c - nc \end{cases}$$ in bounded domains $\Omega\subset\mathbb{R}^N$, $N\ge 1$, under no-flux boundary conditions for $n$ and the physically meaningful condition $$\partial_{\nu} c = (\gamma-c)g$$ on $c$, with given parameter $\gamma>0$ and $g\in C^{1+\beta}(\Omega)$ satisfying $g\ge 0$, $g \not\equiv 0$ on $\partial \Omega$. We prove existence and uniqueness of solutions for any given mass $\int_\Omega n > 0$. These solutions are non-constant

Lack of smoothing for bounded solutions of a semilinear parabolic equation

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as $t\to\infty$

Boundedness of solutions to a virus infection model with saturated chemotaxis

We show global existence and boundedness of classical solutions to a virus infection model with chemotaxis in bounded smooth domains of arbitrary dimension and for any sufficiently regular nonnegative initial data and homogeneous Neumann boundary conditions. More precisely, the system considered is $$\begin{cases}\begin{split} & u_t=\Delta u - \nabla\cdot(\frac{u}{(1+u)^{\alpha}}\nabla v) - uw + \kappa - u, \\ & v_t=\Delta v + uw - v, \\ & w_t=\Delta w - w + v, \end{split}\end{cases}$$ with $\kappa\ge 0$, and solvability and boundedness of the solution are shown under the condition that \[ \begin{cases} \alpha > \frac 12 + \frac{n^2}{6n+4}, &\text{if } \quad 1 \leq n \leq 4 \\ \alpha > \frac {n}4, &\text{if } \quad n \geq 5. \end{cases} \

Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption

Assuming that $0<\chi<\sqrt{\frac{2}n}$, $\kappa\ge 0$ and $\mu>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing $$\begin{split} u_t &= \Delta u - \chi \nabla\cdot ( \frac{u}{v} \nabla v ) + \kappa u -\mu u^2\\ v_t &= \Delta v - u v \end{split}$$ in a bounded domain $\Omega \subset \mathbb{R}^n$, with homogeneous Neumann boundary conditions and for widely arbitrary positive initial data. In the spatially one-dimensional setting, we prove global existence and, moreover, boundedness of the solution for any $\chi>0$, $\mu>0$, $\kappa\ge 0$.Comment: 25 page

On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms

Introducing a suitable solution concept, we show that in bounded smooth domains $\Omega\subset \mathbb{R}^n$, $n\ge 1$, the initial boundary value problem for the chemotaxis system \begin{align*} u_t&=\Delta u -\chi\nabla\cdot\left(\frac{u}{v}\nabla v\right)+\kappa u -\mu u^2,\\ v_t&=\Delta v -uv, \end{align*} with homogeneous Neumann boundary conditions and widely arbitrary initial data has a generalized global solution for any $\mu, \kappa, \chi >0$