NPDE Lecture 4 Discussion

IV. Nonlinear Diffusion in Non-smooth Domains
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In this lecture I am going to talk about nonlinear diffusion equation in general non-smooth domains. In a series of papers I developed the theory of nonlinear diffusion type equations in general non-smooth domains. In this talk I am going to explain the results of my papers [1,2]. A particular motivation for this works arises from the problem about the evolution of interfaces in problems for porous medium equation. Special interest concerns the cases when support of the initial data contains a corner or cusp singularity at some points. What about the movement of these kind of singularities along the interface? To solve this problem, it is important at the first stage to develop general theory of boundary-value problems in non-cylindrical domains with boundary surfaces which has the same kind of behaviour as the interface. In many cases this may be non-smooth and characteristic. It should be mentioned that in the one-dimensional case Dirichlet and Cauchy-Dirichlet problems for the reaction-diffusion equations in irregular domains were studied in [3,4]. Primarily applying this theory a complete description of the evolution of interfaces was presented in [5,6].

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NPDE Lecture 2 Discussion

II. Weak Solutions and Uniqueness
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Weak Solutions

Let me remind you what I did in a previous lecture. I considered the problem
\begin{align}
u_{t}-\text{div}\big( u^{\sigma} \nabla u\big) &=0,~x \in\R^{N},~t>0; \ \sigma > 0,\label{eq:inst-pme-1}\\
\int_{\R^{N}}u(x,t)\,dx &=1,~t > 0\label{eq:inst-pme-2}\\
u(x,0) = \delta(x)\label{eq:inst-pme-3}
\end{align}
and constructed instantaneous point-source solution, also called the Zeldovich-Kompaneets-Barenblatt (ZKB) solution.
\begin{equation} u_*(x,t)=t^{-\frac{N}{2+N\sigma}}\left[\frac{\sigma}{2(2+N\sigma)} \left( \eta_{0}^{2}-\frac{|x|^{2}}{t^{\frac{2}{2+N\sigma}}}\right)_{+} \right]^{\frac{1}{\sigma}}\end{equation}
Note that the nonlinear diffusion equation is invariant under translation of time and space coordinate, and the solution to the problem
\[ \begin{cases}
u_{t}=\text{div}\big(u^{\sigma }u\big),~&(x,t) \in \R^{N}\times \R_{+}\\
u(x,0)=u_*(x,T),~&x \in \R^{N}
\end{cases}\]
is exactly $u_*(x,t+T)$, which has compact support for all $t\geq 0$.

However, several important questions are left open. First of all, ZKB solution is not a classical solution: it is not even differentiable on the boundary of the support. First important question which we need to answer is the following: In what sense is $u_*$ actually a solution of the problem? Hence, we need to define the notion of the weak solution. Second important question we need to answer is the following: May be there is a smooth solution of the same problem with different properties, and ZKB solution is just physically irrelevant mathematical example. Main goal of this lecture is to answer these important questions.
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NPDE Localization Effect in Diffusion-Absorption Equation

Reaction-diffusion equations can exhibit a phenomenon known as space localization: not only do they exhibit finite speed of propagation, but heat can be completely contained in a compact region for all time under certain conditions. This is a physically pertinent phenomena as well, having been observed in plasmas, but not encountered in solutions to the linear heat equation.

Consider the reaction-diffusion equation
\begin{equation} u_{t}=\text{div} \big( u^{\sigma}\nabla u\big)-\gamma u,\quad u \in \R^{N},~t>0\label{eq:pme-loc-1}\end{equation}
where $\sigma >0$, $\gamma>0$. That means nonlinear diffusion is accompanied with linear absorption.
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