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.

Let
\[ u(x,t)=e^{-\gamma t}v(x,t)\]
Then
\begin{gather*}
u_{t}=e^{-\gamma t}v_{t} – \gamma e^{-\gamma t}v\\
\end{gather*}
Substituting this to \eqref{eq:pme-loc-1} gives
\begin{gather*}
e^{-\gamma t}v_{t} – \gamma e^{-\gamma t}v = \text{div}\left( e^{-\gamma \sigma t} v^{\sigma} \nabla\big(e^{-\gamma t}v \big)\right)-\gamma e^{-\gamma t}v
\end{gather*}
So
\begin{gather*}
e^{\gamma\sigma t} v_{t}=\text{div}\big( v^{\sigma}\nabla v\big)
\end{gather*}
In order to remove the coefficient $e^{\gamma \sigma t}$ introduce new time variable
\begin{gather*}
\tau=\tau(t),\quad v(x,t) = w(x,\tau)
\end{gather*}
It follows that
\begin{gather*}
e^{\gamma \sigma t}\tau’ w_{\tau} = \text{div}\big( w^{\sigma}\nabla w\big)
\end{gather*}
Setting
\begin{gather*}
\tau'(t)=e^{-\gamma \sigma t}, \ \tau(0)=0\Implies \tau(t)=\frac{1}{\gamma \sigma} \left[ 1-e^{-\gamma \sigma t}\right]
\end{gather*}
Then $\tau\colon [0,+\infty) \to [0,1/\gamma \sigma)$, and $w$ solves
\begin{gather*}
w_{\tau} = \text{div}\big( w^{\sigma}\nabla w\big), \ x\in \R^N, \ 0<\tau<\frac{1}{\gamma \sigma} \end{gather*} Choose as $w$ the ZKB solution constructed in a Lecture 1: \begin{align*} w(x,\tau)&=u_*(x,1+\tau)\\ &=\left( 1+\tau\right)^{-\frac{N}{2+N\sigma}} \left[ \frac{\sigma}{2(2+N\sigma)}\left( \eta_{0}^{2} - \frac{\lnorm{x}^{2}}{\lnorm{1+\tau}^{\frac{2}{2+N\sigma}}}\right)_{+}\right]^{\frac{1}{\sigma}} \end{align*} Transforming this solution back to $u(x,t)$ gives \begin{align*} u(x,t)&=e^{-\gamma t} \left[ 1+\tau\right]^{-\frac{N}{2+N\sigma}}\left[ \frac{\sigma}{2(2+N\sigma)}\left( \eta_{0}^{2} - \frac{\lnorm{x}^{2}}{\lnorm{1+\tau}^{\frac{2}{2+N\sigma}}}\right)_{+}\right]^{\frac{1}{\sigma}} \end{align*} The support of this solution is \begin{align*} \text{supp} u&=\bk{(x,t):|x|<\eta_{0}\left( 1+\tau(t)\right)^{\frac{1}{2+N\sigma}}}\\ &=\bk{(x,t):\vert x\vert \lt x_{*}(t)=\eta_{0}\left[ 1+\frac{1-e^{-\gamma \sigma t}}{\gamma \sigma}\right]^{\frac{1}{2+N\sigma}}} \end{align*} We calculate that the free boundary has asymptotic behavior \[ L=\lim_{t\to+\infty}x_{*}(t)=\eta_{0}\left( 1+\frac{1}{\gamma \sigma}\right)^{\frac{1}{2+N\sigma}}<\infty\] That is, the solution $u(x,t)$ for all $t\geq 0$ is localized in a ball of radius $L$ in $\R^N$. In fact, without the absorption term $-\gamma u$, localization phenomena doesn't happen, and diffusion eventually covers whole space when $t \to +\infty$. By the way, one can see that $L\to +\infty$ as $\gamma \to 0$. Space localization phenomena was first observed in

  1. L.K.Martinson and K.B.Pavlov, Unsteady shear flows of a conducting fluid with a rheological power law, Magnetohydrodynamics (USSR) 7 (1971), no.~2, 50-58.
  2. L.K.Martinson and K.B.Pavlov, The problem of the three-dimensional localization of thermal perturabations in the theory of nonlinear heat conduction, USSR Computational Mathematics and Mathematical Physics 12, 4(1972), 261-268.

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