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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# Nonlinear Partial Differential Equations

Partial differential equations (PDEs) are central to mathematics, whether pure or applied. Kanken Big They arise in mathematical models of real world problems, where dependent variables vary continuously as functions of several independent variables, usually space and time. nike free trainer PDEs arising in a majority of real world applications are nonlinear. Ugg 2017 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. Compra Mochilas Kanken 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. timberland soldes 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.

# NPDE Lecture 4 Discussion

IV. Nonlinear Diffusion in Non-smooth Domains

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# 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 \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. Continue reading NPDE Lecture 2 Discussion

# 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. nike requin 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 $\sigma >0$, $\gamma>0$. nike air max 90 That means nonlinear diffusion is accompanied with linear absorption. adidas stan smith Continue reading NPDE Localization Effect in Diffusion-Absorption Equation