On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions

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 $p<q$ and in $N$-dimensional settings, we establish a \textit{global existence criterion} for $u$. \end{itemize

Constant sign and nodal solutions for nonhomogeneous Robin boundary value problems with asymmetric reactions

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 page

Reaction-diffusion problems under non-local boundary conditions with blow-up solutions

This paper deals with blow-up solutions to a class of reaction-diffusion equations under non-local boundary conditions. We prove that under certain conditions on the data the blow-up will occur at some finite time and when the blow-up does occur, lower and upper bounds are derived

On explicit lower bounds and blow-up times in a model of chemotaxis

Abstract. This paper is concerned with a parabolic Keller-Segel system in R^n, with n = 2 and 3, under Neumann boundary conditions on the boundary. Firstly important theoretical and general results dealing with lower bounds for blow-up time estimates are summarized and analyzed. Secondly, a resolution method is proposed and used to both compute the real blow-up times of such unbounded solutions and analyze and discuss some of their properties

Behavior in time of solutions of a Keller–Segel system with flux limitation and source term

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), &amp; \\[2mm] 0= \Delta v -m(t)+ u , \quad \int_{\Omega}v \,dx=0, &amp; \\[2mm] u(x,0)= u_0(x), &amp; \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

Blow-up and Decay Bounds in Hamilton-Jacobi-type Problems

Abstract In this paper, we investigate a class of nonlinear parabolic problems, known as viscous Hamilton-Jacobi-type problems. We establish conditions on data sufficient to insure that blow-up occurs in finite time. Moreover, conditions on data and geometry of the spatial domain are derived, ensuring the solution to exist for all time with exponential decay

The radiosensitizing effect of Ku70/80 knockdown in MCF10A cells irradiated with X-rays and p(66)+Be(40) neutrons

Background: A better understanding of the underlying mechanisms of DNA repair after low-and high-LET radiations represents a research priority aimed at improving the outcome of clinical radiotherapy. To date however, our knowledge regarding the importance of DNA DSB repair proteins and mechanisms in the response of human cells to high-LET radiation, is far from being complete. Methods: We investigated the radiosensitizing effect after interfering with the DNA repair capacity in a human mammary epithelial cell line (MCF10A) by lentiviral-mediated RNA interference (RNAi) of the Ku70 protein, a key-element of the non-homologous end-joining (NHEJ) pathway. Following irradiation of control and Ku-deficient cell lines with either 6 MV X-rays or p(66)+Be(40) neutrons, cellular radiosensitivity testing was performed using a crystal violet cell proliferation assay. Chromosomal radiosensitivity was evaluated using the micronucleus (MN) assay. Results: RNAi of Ku70 caused downregulation of both the Ku70 and the Ku80 proteins. This downregulation sensitized cells to both X-rays and neutrons. Comparable dose modifying factors (DMFs) for X-rays and neutrons of 1.62 and 1.52 respectively were obtained with the cell proliferation assay, which points to the similar involvement of the Ku heterodimer in the cellular response to both types of radiation beams. After using the MN assay to evaluate chromosomal radiosensitivity, the obtained DMFs for X-ray doses of 2 and 4 Gy were 2.95 and 2.66 respectively. After neutron irradiation, the DMFs for doses of 1 and 2 Gy were 3.36 and 2.82 respectively. The fact that DMFs are in the same range for X-rays and neutrons confirms a similar importance of the NHEJ pathway and the Ku heterodimer for repairing DNA damage induced by both X-rays and p(66)+Be(40) neutrons. Conclusions: Interfering with the NHEJ pathway enhanced the radiosensitivity of human MCF10A cells to low-LET Xrays and high-LET neutrons, pointing to the importance of the Ku heterodimer for repairing damage induced by both types of radiation. Further research using other high-LET radiation sources is however needed to unravel the involvement of DNA double strand break repair pathways and proteins in the cellular response of human cells to high-LET radiation