Related papers: Interaction between nonlinear diffusion and geomet…
We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, $\partial _t u=J*u-u$, where $J$ is a smooth, radially symmetric kernel with support $B_d(0)\subset\mathbb{R}^2$. The problem is set in an…
We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory,…
The interface problem for the linear Schr\"odinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the…
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial \Omega$. In this memoir, we consider issues of non-existence, existence, and multiplicity of variational solutions in $H_{1,0}^2(\Omega)$ for the…
This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size $\varepsilon$, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the…
We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some…
We consider the effective surface motion of a particle that intermittently unbinds from a planar surface and performs bulk excursions. Based on a random walk approach we derive the diffusion equations for surface and bulk diffusion…
We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set $\Omega \subseteq \mathbb{R}^n$. Specifically, we show that if…
We study nonnegative solutions to the Cauchy problem for the Fractional Fast Diffusion Equation on a suitable class of connected, noncompact Riemannian manifolds. This parabolic equation is both singular and nonlocal: the diffusion is…
Let $\Omega \subset \mathbb{R}^n$, for $n \geq 2$, be a bounded $C^2$ domain. Let $q \in L^1_{loc} (\Omega)$ with $q \geq 0$. We give necessary conditions and matching sufficient conditions, which differ only in the constants involved, for…
We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the…
Let $\Omega$ be a smooth bounded domain in $\R^n$, $n\ge 3$, $0<m\le\frac{n-2}{n}$, $a_1,a_2,..., a_{i_0}\in\Omega$, $\delta_0=\min_{1\le i\le i_0}{dist }(a_i,\1\Omega)$ and let…
Fix $R>0$ and let $B_R$ denote the ball of radius $R$ centered at the origin in $R^d$, $d\ge2$. Let $D\subset B_R$ be an open set with smooth boundary and such that $R^d-\bar D$ is connected, and let $$…
The current paper considers the boundedness of solutions to the following quasilinear Keller-Segel model (with logistic source) $$\left\{\begin{array}{ll} u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\mu (u-u^2),\quad x\in…
In this paper we present in one-dimensional space a numerical solution of a partial differential equation of fractional order. This equation describes a process of anomalous diffusion. The process arises from the interactions within the…
Uchaikin suggested a mathematical model of an anomalous diffusion in a space was suggested. This model origins in an investigation of processes in complex systems with variable structure: glasses, liquid crystals, biopolymers, proteins and…
In this article, we study domains $\Omega \subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ \Delta u + f(u,|\nabla u|)=0 \quad \text{in } \Omega, $$ subject to the boundary conditions $u=0$ on…
The Smoluchowski diffusion equation describes diffusion in the presence of external forces. Studying the mechanical response of soft materials to linear forces, such as shear, results in a boundary value problem involving an…
This paper is devoted to the anomalous diffusion limit of kinetic equations with a fractional Fokker-Planck collision operator in a spatially bounded domain. We consider two boundary conditions at the kinetic scale: absorption and specular…
The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll}…