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

Boundedness in a fully parabolic chemotaxis system with nonlinear diffusion and sensitivity, and logistic source

In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{ t}=\nabla \cdot ((u+1)^{m-1} \nabla u-(u+1)^\alpha \chi(v)\nabla v) + ku-\mu u^2 & x\in \Omega, t>0, \\ v_{t} = \Delta v-vu & x\in \Omega, t>0,\\ \end{cases} \end{equation*} $\Omega$ being a bounded and smooth domain of $\mathbb{R}^n$, $n\geq 1$, and where $m,k \in \mathbb{R}$, $\mu>0$ and $\alpha < \frac{m+1}{2}$. For any $v\geq 0$ the chemotactic sensitivity function is assumed to behave as the prototype $\chi(v) = \frac{\chi_0}{(1+av)^2}$, with $a\geq 0$ and $\chi_0>0$. We prove that for nonnegative and sufficiently regular initial data $u(x,0)$ and $v(x,0),$ the corresponding initial-boundary value problem admits a global bounded classical solution provided $\mu$ is large enough

Zu einer Ästhetik der Schwelle. Philosophische Reflexionen am Beispiel vom Musiktheater im Revier

The essay develops an aesthetic of the threshold from an analysis of the artistic collaboration between Werner Ruhnau and Yves Klein for the realisation of the Musiktheater im Revier in Gelsenkirchen. The threshold is a figure of thought that stands for the artistic choice to create zones of interaction between work and spectator, in which exchanges of roles and production of meaning take place. The aesthetics of the threshold in Ruhnau manifests itself through the architectural design of playful theatrical spaces in which the theatre audience can stage new identities. In Klein, the threshold becomes a meditative space, through which the subject trains his faculties of feeling and making sense. In the “Verfransung” (Adorno) between the strategies of Ruhnau and Klein, the theatre becomes the threshold for an interaction between public space and sensibility.The essay develops an aesthetic of the threshold from an analysis of the artistic collaboration between Werner Ruhnau and Yves Klein for the realisation of the Musiktheater im Revier in Gelsenkirchen. The threshold is a figure of thought that stands for the artistic choice to create zones of interaction between work and spectator, in which exchanges of roles and production of meaning take place. The aesthetics of the threshold in Ruhnau manifests itself through the architectural design of playful theatrical spaces in which the theatre audience can stage new identities. In Klein, the threshold becomes a meditative space, through which the subject trains his faculties of feeling and making sense. In the “Verfransung” (Adorno) between the strategies of Ruhnau and Klein, the theatre becomes the threshold for an interaction between public space and sensibility

Mathematical modeling on gas turbine blades/vanes under variable convective and radiative heat flux with tentative different laws of cooling

In the last twenty years the modeling of heat transfer on gas turbine cascades has been based on computational fluid dynamic and turbulence modeling at sonic transition. The method is called Conjugate Flow and Heat Transfer (CHT). The quest for higher Turbine Inlet Temperature (TIT) to increase electrical efficiency makes radiative transfer the more and more effective in the leading edge and suction/ pressure sides. Calculation of its amount and transfer towards surface are therefore needed. In this paper we decouple convection and radiation load, the first assumed from convective heat transfer data and the second by means of emissivity charts and analytical fits of heteropolar species as CO2 and H2O. Then we propose to solve the temperature profile in the blade through a quasi-two-dimensional power balance in the form of a second order partial differential equation which includes radiation and convection. Real cascades are cooled internally trough cool compressed air, so that we include in the power balance the effect of a heat sink or law of cooling that is up to the designer to test in order to reduce the thermal gradients and material temperature. The problem is numerically solved by means of the Finite Element Method (FEM) and, subsequently, some numerical simulations are also presented