We consider the heat flux through a domain with subregions in which the thermal capacity approaches zero. In these subregions the parabolic heat equation degenerates to an elliptic one. We show the well-posedness of such parabolic-elliptic differential equations for general non-negative L∞-capacities and study the continuity of the solutions with respect to the capacity, thus giving a rigorous ...

Let S (respectively, S 0) be a nite subset of a compact connected Riemann surface X (respectively, X 0) of genus at least two. Let M (respectively, M 0) denote a moduli space of parabolic stable bundles of rank two over X (respectively, X 0) with xed determinant of degree one, having a nontrivial quasi-parabolic structure over each point of S (respectively, S 0), and of parabolic degree less th...

Let $\mathbf{G}$ be a connected reductive group defined over locally compact non-archimedean field $F$, let $\mathbf{P}$ parabolic subgroup with Levi $\mathbf{M}$ and compatible pro-$p$ Iwahori of $G := \mathbf{G}(F)$. $R$ commutative unital ring. We introduce the Iwahori--Hecke $R$-algebra $\mathcal{H}_R(P)$ $P \mathbf{P}(F)$ construct two morphisms $\Theta^P_M\colon \mathcal{H}_R(P)\to \mathc...

In this paper, we show that the center of every Coxeter group is finite and isomorphic to (Z2) n for some n ≥ 0. Moreover, for a Coxeter system (W, S), we prove that Z(W ) = Z(W S\S̃) and Z(W S̃ ) = 1, where Z(W ) is the center of the Coxeter group W and S̃ is the subset of S such that the parabolic subgroup W S̃ is the essential parabolic subgroup of (W, S) (i.e. W S̃ is the minimum parabolic subgr...

In this paper, we study controllability for a parabolic system of chemotaxis. With one control only, the local exact controllability to positive trajectory of the system is obtained by applying Kakutani’s fixed point theorem and the null controllability of associated linearized parabolic system. The positivity of the state is shown to be remained in the state space. The control function is show...

In this paper is presented the problem of optimizing a functional over a Pareto control set associated with a convex multiobjective control problem in Hilbert spaces, namely parabolic system. This approach generalizes for this setting some results obtained in finite dimensions. Some examples are presented. General optimality results are obtained, and a special attention is paid to the linear-qu...

A Gauss–Galerkin finite-difference method is proposed for the numerical solution of a class of linear, singular parabolic partial differential equations in two space dimensions. The method generalizes a Gauss– Galerkin method previously used for treating similar singular parabolic partial differential equations in one space dimension. Two test problems are studied and the numerical results are ...

We prove Calderón-Zygmund estimates for a class of parabolic problems whose model is the non-homogeneous parabolic p(x, t)-Laplacian equation ∂tu − div ( |Du|p(x,t)−2Du ) = f − div ( |F|p(x,t)−2F ) . More precisely, we will show that the spatial gradient Du is as integrable as the inhomogeneities f and F, i.e. |F|p(x,t), | f | γ1 γ1−1 ∈ L loc ⇒ |F| p(x,t) ∈ L loc for any q > 1, where γ1 is the ...

We review an emerging application field to parabolic partial differential equations (PDEs), that’s economic growth theory. After a short presentation of concrete applications, we highlight the peculiarities of optimal control problems of parabolic PDEs with infinite time horizons. In particular, the heuristic application of the maximum principle to the latter leads to single out a serious illpo...