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Register a new userOn the existence of solutions for a drift-diffusion system arising in corrosion modelling
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In this paper, we consider a drift-diffusion system describing the corrosion of an iron based alloy in a nuclear waste repository. In comparison with the classical drift-diffusion system arising in the modeling of semiconductor devices, the originality of the corrosion model lies in the boundary conditions which are of Robin type and induce an additional coupling between the equations. We prove the existence of a weak solution by passing to the limit on a sequence of approximate solutions given by a semi-discretization in time.
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We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on $p$ and $m$ in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincare inequality hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in ${mathbb R}^n$.
Let $Omega$ be a smooth bounded domain in $R^n$, $nge 3$, $0<mlefrac{n-2}{n}$, $a_1,a_2,..., a_{i_0}inOmega$, $delta_0=min_{1le ile i_0}{dist }(a_i,1Omega)$ and let $Omega_{delta}=Omegasetminuscup_{i=1}^{i_0}B_{delta}(a_i)$ and $hat{Omega}=Omegasetminus{a_1,...,a_{i_0}}$. For any $0<delta<delta_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=Delta u^m$ in $Omega_{delta}times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $hat{Omega}times (0,T)$ for some $T>0$ that blow-up at the points $a_1,..., a_{i_0}$.
In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key element of our proof is the control of a particular defect measure associated to the pressure which avoids the use of the eective ux. Using this new tool, we solve an open problem namely global existence of solutions {`a} la Leray for such a system without assuming any restriction on the anisotropy amplitude. It provides a exible and natural way to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.
We study the local in time existence of a regular solution of a nonlinear parabolic backward-forward system arising from the theory of Mean-Field Games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.
We study the existence of multi-bubble solutions for the following skew-symmetric Chern--Simons system begin{equation}label{e051} left{ begin{split} &Delta u_1+frac{1}{varepsilon^2}e^{u_2}(1-e^{u_1})=4pisum_{i=1}^{2k}delta_{p_{1,i}} &Delta u_2+frac{1}{varepsilon^2}e^{u_1}(1-e^{u_2})=4pisum_{i=1}^{2k}delta_{p_{2,i}} end{split} text{ in }quad Omegaright., end{equation} where $kgeq 1$ and $Omega$ is a flat tours in $mathbb{R}^2$. It continues the joint work with Zhangcite{HZ-2015}, where we obtained the necessary conditions for the existence of bubbling solutions of Liouville type. Under nearly necessary conditions(see Theorem ref{main-thm}), we show that there exist a sequence of solutions $(u_{1,varepsilon}, u_{2,varepsilon})$ to eqref{e051} such that $u_{1,varepsilon}$ and $u_{2,varepsilon}$ blow up simultaneously at $k$ points in $Omega$ as $varepsilonto 0$.
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