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In this work we study the degenerate diffusion equation $partial_{t}=x^{alpha}aleft(xright)partial_{x}^{2}+bleft(xright)partial_{x}$ for $left(x,tright)inleft(0,inftyright)^{2}$, equipped with a Cauchy initial data and the Dirichlet boundary condition at $0$. We assume that the order of degeneracy at 0 of the diffusion operator is $alphainleft(0,2right)$, and both $aleft(xright)$ and $bleft(xright)$ are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution $pleft(x,y,tright)$ and prove several properties for $pleft(x,y,tright)$; by conducting a localization procedure, we obtain an approximation for $pleft(x,y,tright)$ for $x,y$ in a neighborhood of 0 and $t$ sufficiently small, where the error estimates only rely on the local bounds of $aleft(xright)$ and $bleft(xright)$ (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of $alpha=1$. Our work extends part of the existing results to cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probability view (e.g., wellposedness of stochastic differential equations).
In this paper, we present strong numerical evidences that the $3$D incompressible axisymmetric Navier-Stokes equations with degenerate diffusion coefficients and smooth initial data of finite energy develop a potential finite time locally self-simila
We consider properties of second-order operators $H = -sum^d_{i,j=1} partial_i , c_{ij} , partial_j$ on $Ri^d$ with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) geq 0$ almost everywhere, but allow for the possibility t
The goal of this work is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a
We derive a diffusion approximation for the kinetic Vlasov-Fokker-Planck equation in bounded spatial domains with specular reflection type boundary conditions. The method of proof involves the construction of a particular class of test functions to b
In this paper, we study the consumption-chemotaxis-Stokes model with porous medium slow diffusion in a three dimensional bounded domain with zero-flux boundary conditions and no-slip boundary condition. In recent ten years, many efforts have been mad