VI. Evolution of Interfaces for the Reaction-Diffusion Equations – Part 2

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# Category Archives: NPDE

Nonlinear PDE Research Topic

# NPDE Lecture 5 Discussion

# NPDE Lecture 1 Discussion

I. Nonlinear Diffusion Equation (Introduction)

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The importance of the nonlinear PDEs is associated with the fact that they reflect more precise nonlinear laws of the nature, while linear PDEs were derived from the linearized versions of the nonlinear laws. Despite the great advances in the theory of nonlinear PDEs it is far from being complete. It is remarkable that almost every major nonlinear PDE has its own personality and requires unique approach. However, looking to current state of art in the theory of nonlinear PDEs, one can observe a similarity with the classical linear theory in the following sense: there are some major individual nonlinear PDEs, and analysis and understanding of those PDEs is a key towards the general theory of a broad class of nonlinear PDEs. I am going to concentrate on one of those key nonlinear PDEs called the nonlinear diffusion equation, which is a generalization of the classical heat/diffusion equation.

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# Nonlinear Partial Differential Equations

Partial differential equations (PDEs) are central to mathematics, whether pure or applied. They arise in mathematical models of real world problems, where dependent variables vary continuously as functions of several independent variables, usually space and time. PDEs arising in a majority of real world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs one can

observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear reaction-diffusion-convection equation:

\[u_t=(u^m)_{xx}+bu^{\beta}+c(u^{\gamma})_x,\]

where \[u=u(x,t),x\in \mathbb{R}, t>0, b,c \in \mathbb{R}, m>0, \beta>0, \gamma>0\]

It arises, for instance, in the theory of gas flow through a porous medium, heat radiation in plasmas, spatial spread of biological populations, diffusion of particles in plasma etc.

In summer session of 2014 students will pursue research on the open problem of evolution of interfaces

of solutions to the reaction-diffusion-convection equations.

Full solution of this problem for the reaction-diffusion equation is given in the following two papers by using significantly the theory of reaction-diffusion equations in general non-smooth domains developed in my JDE paper.

- U. G. Abdulla and J. R. King, Interface development and local solutions to reaction-diffusion equations, SIAM J. Math. Anal., 32, 2, 2000, 235-260.
- U. G. Abdulla, Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption, Nonlinear Analysis, 50, 4, 2002, 541-560.
- U. G. Abdulla, Reaction-diffusion in irregular domains, J. Differential Equations, 164, 2000, 321-354.

Introductory lectures on this topic will be posted soon.

# NPDE Lecture 4 Discussion

IV. Nonlinear Diffusion in Non-smooth Domains

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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].

# NPDE Lecture 3 Discussion

III. Energy Estimates and Existence

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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

[latex]

\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}

[/latex]

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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