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In this paper we consider radially symmetric solutions of the following parabolic--elliptic cross-diffusion system \begin{equation*} \begin{cases} u_t = \Delta u - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), & \\[2mm] 0= \Delta v -m(t)+ u , \quad \int_{\Omega}v \,dx=0, & \\[2mm] u(x,0)= u_0(x), & \end{cases} \end{equation*} in $\Omega \times (0,\infty)$, with $\Omega$ a ball in $\mathbb{R}^N$, $N\geq 3$, under homogeneous Neumann boundary conditions, where $g(u)= \lambda u - \mu u^k$ , $\lambda >0, \ \mu >0$, and $ k >1$, $f(|\nabla v|^2 )= k_f(1+ |\nabla v|^2)^{-\alpha}$, $\alpha>0$, which describes gradient-dependent limitation of cross diffusion fluxes. The function $m(t)$ is the time dependent spatial mean of $u(x,t)$ i.e. $m(t) := \frac 1 {|\Omega|} \int_{\Omega} u(x,t) \,dx$. Under smallness conditions on $\alpha$ and $k$, we prove that the solution $u(x,t)$ blows up in $L^{\infty}$-norm at finite time $T_{max}$ and for some $p>1$ it blows up also in $L^p$-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on $\alpha$ or $k$, we prove that the solution is global and bounded in time., Comment: 32 pages. arXiv admin note: text overlap with arXiv:2201.08716

This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u \nabla v) + \xi \nabla\cdot (u \nabla w) + \lambda u - \mu u^k, \quad &x \in \Omega,\ t>0,\\[1.05mm] 0= \Delta v + \alpha u - \beta v, \quad &x \in \Omega,\ t>0,\\[1.05mm] 0= \Delta w + \gamma u - \delta w, \quad &x \in \Omega,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions, in a ball $\Omega \subset \mathbb{R}^n$ ($n \ge 3$), with constant parameters $\lambda \in \mathbb{R}$, $k>1$, $\mu, \chi, \xi, \alpha, \beta, \gamma, \delta>0$. Blow-up phenomena in the system have been well investigated in the case $\lambda=\mu=0$, whereas the attraction-repulsion chemotaxis system with logistic degradation has been not studied. Under the condition that $k>1$ is close to $1$, this paper ensures a solution which blows up in $L^\infty$-norm and $L^\sigma$-norm with some $\sigma>1$ for some nonnegative initial data. Moreover, a lower bound of blow-up time is derived.

In this paper we analyze the porous medium equation \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} %\begin{cases} u_t=\Delta u^m + a\io u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad \textrm{in}\quad \Omega \times I,%\\ %u_\nu-g(u)=0 & \textrm{on}\; \partial \Omega, t>0,\\ %u({\bf x},0)=u_0({\bf x})&{\bf x} \in \Omega,\\ %\end{cases} \end{equation} where $\Omega$ is a bounded and smooth domain of $\R^N$, with $N\geq 1$, and $I= [0,t^*)$ is the maximal interval of existence for $u$. The constants $a,b,c$ are positive, $m,p,q$ proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of $u$. Under some hypothesis on the data, including intrinsic relations between $m,p$ and $q$, and assuming that for some positive and sufficiently regular function $u_0(\nx)$ the Initial Boundary Value Problem (IBVP) associated to \eqref{ProblemAbstract} possesses a positive classical solution $u=u(\nx,t)$ on $\Omega \times I$: \begin{itemize} \item [$\triangleright$] when $p>q$ and in 2- and 3-dimensional domains, we determine a \textit{lower bound of} $t^*$ for those $u$ becoming unbounded in $L^{m(p-1)}(\Omega)$ at such $t^*$; \item [$\triangleright$] when $p

This paper deals with unbounded solutions to the following zero--flux chemotaxis system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} % about u u_t=\nabla \cdot [(u+\alpha)^{m_1-1} \nabla u-\chi u(u+\alpha)^{m_2-2} \nabla v] & (x,t) \in \Omega \times (0,T_{max}), \\[1mm] % about v 0=\Delta v-M+u & (x,t) \in \Omega \times (0,T_{max}), \end{cases} \end{equation} where $\alpha>0$, $\Omega$ is a smooth and bounded domain of $\mathbb{R}^n$, with $n\geq 1$, $t\in (0, T_{max})$, where $T_{max}$ the blow-up time, and $m_1,m_2$ real numbers. Given a sufficiently smooth initial data $u_0:=u(x,0)\geq 0$ and set $M:=\frac{1}{|\Omega|}\int_{\Omega}u_0(x)\,dx$, from the literature it is known that under a proper interplay between the above parameters $m_1,m_2$ and the extra condition $\int_\Omega v(x,t)dx=0$, system \eqref{ProblemAbstract} possesses for any $\chi>0$ a unique classical solution which becomes unbounded at $t\nearrow T_{max}$. In this investigation we first show that for $p_0>\frac{n}{2}(m_2-m_1)$ any blowing up classical solution in $L^\infty(\Omega)$--norm blows up also in $L^{p_0}(\Omega)$--norm. Then we estimate the blow--up time $T_{max}$ providing a lower bound $T$.

We investigate a PDE-constrained optimization problem, with an intuitive interpretation in terms of the design of robust membranes made out of an arbitrary number of different materials. We prove existence and uniqueness of solutions for general smooth bounded domains, and derive a symmetry result for radial ones. We strengthen our analysis by proving that, for this particular problem, there are no non-global local optima. When the membrane is made out of two materials, the problem reduces to a shape optimization problem. We lay the preliminary foundation for computable analysis of this type of problem by proving stability of solutions with respect to some of the parameters involved., Comment: 19 pages, 1 figure

We study a nonlinear, nonhomogeneous elliptic equation with an asymmetric reaction term depending on a positive parameter, coupled with Robin boundary conditions. Under appropriate hypotheses on both the leading differential operator and the reaction, we prove that, if the parameter is small enough, the problem admits at least four nontrivial solutions: two of such solutions are positive, one is negative, and one is sign-changing. Our approach is variational, based on critical point theory, Morse theory, and truncation techniques., Comment: 22 pages

By means of nonsmooth critical point theory, we prove existence of three weak solutions for an ordinary differential inclusion of Sturm-Liouville type involving a general set-valued reaction term depending on a parameter, and coupled with mixed boundary conditions. As an application, we give a multiplicity result for ordinary differential equations involving discontinuous nonlinearities., Comment: 11 pages