Diffusive stability

Diffusive stability is a property of solutions to reaction diffusion equations . Under certain conditions these, the typical diffusion property that their norm for converging to 0 while the norm is simultaneously limited. In particular, the constant zero solution is asymptotically stable . This clearly means for a solution that models the concentration of a substance in space as a function of time that the concentration of the substance is evenly distributed over time, while the total concentration itself is always limited. This property of solutions is also known as diffusive stability . ${\displaystyle L^{\infty }}$ ${\displaystyle t\to \infty }$ ${\displaystyle L^{1}}$

Behavior with linear diffusion equation

For the linear diffusion equation in a space dimension

Solution of the linear diffusion equation with initial condition . ${\displaystyle t=1}$ ${\displaystyle u_{0}(x)=e^{-x^{2}}}$

${\displaystyle {\frac {\partial }{\partial t}}u={\frac {\partial ^{2}}{\partial x^{2}}}u}$ With ${\displaystyle x\in \mathbb {R} ,t\in [0,\infty )}$

with the initial condition the solution is given by the convolution of the initial condition with the fundamental solution ${\displaystyle u(x,0)=u_{0}(x)}$

${\displaystyle H(x,t)={\frac {1}{\sqrt {4\pi t}}}e^{-{\frac {x^{2}}{4t}}}}$.

Be now . ${\displaystyle u_{0}(\cdot )\in L^{1}}$

Then the solution can be written generally as

${\displaystyle u(x,t)=(H(\cdot ,t)*u_{0}(\cdot ))(x)=\int _{-\infty }^{\infty }H(x-y,t)u_{0}(y)\,dy}$.

According to Young's inequality for convolution it follows on the one hand

${\displaystyle \|u(\cdot ,t)\|_{L^{\infty }}=\left\|\int _{-\infty }^{\infty }H(\cdot -y,t)u_{0}(y)\,dy\right\|_{L^{\infty }}\leq \|H(\cdot ,t)\|_{L^{\infty }}\|u_{0}(\cdot )\|_{L^{1}}}$

Solution of the linear diffusion equation with initial condition . ${\displaystyle t=5}$ ${\displaystyle u_{0}(x)=e^{-x^{2}}}$

on the other hand too

${\displaystyle \|u(\cdot ,t)\|_{L^{1}}=\left\|\int _{-\infty }^{\infty }H(\cdot -y,t)u_{0}(y)\,dy\right\|_{L^{1}}\leq \|H(\cdot ,t)\|_{L^{1}}\|u_{0}(\cdot )\|_{L^{1}}}$.

In addition, the norms of the fundamental solution can be calculated explicitly (see also normal distribution and error integral ).

It applies

${\displaystyle \|H(\cdot ,t)\|_{L^{\infty }}={\frac {1}{\sqrt {2t\pi }}}}$

and

${\displaystyle \|H(\cdot ,t)\|_{L^{1}}={\frac {1}{\sqrt {4\pi t}}}\int _{-\infty }^{\infty }e^{-{\frac {x^{2}}{4t}}}=1}$.

The norm estimates therefore apply overall

Solution of the linear diffusion equation with initial condition . ${\displaystyle t=10}$ ${\displaystyle u_{0}(x)=e^{-x^{2}}}$

${\displaystyle \|u(\cdot ,t)\|_{L^{\infty }}\leq {\frac {1}{\sqrt {4\pi t}}}\|u_{0}(\cdot )\|_{L^{1}}{\overset {t\to \infty }{\to }}0}$

${\displaystyle \|u(\cdot ,t)\|_{L^{1}}\leq \|u_{0}(\cdot )\|_{L^{1}}}$.

These properties correspond exactly to what one would expect from a diffusion, namely that the substance concentration is distributed more and more in the room as it grows , without the total amount of substance increasing. ${\displaystyle u(x,t)}$ ${\displaystyle t}$

Generalization to reaction diffusion equations

Under certain conditions the property of diffusive stability can also be applied to general reaction diffusion equations of the form

${\displaystyle {\frac {\partial }{\partial t}}u={\frac {\partial ^{2}}{\partial x^{2}}}u+f(u)}$ With ${\displaystyle u(0,x)=u_{0}(x)}$

to be generalized.

For example, consider the equation

${\displaystyle {\frac {\partial }{\partial t}}u={\frac {\partial ^{2}}{\partial x^{2}}}u+u^{p}}$

for . The solution to this equation can be written as ${\displaystyle p\geq 1}$

${\displaystyle u(x,t)=(H(\cdot ,t)*u_{0}(\cdot ))(x)+\int _{0}^{t}(H(\cdot ,t-s)*u(\cdot ,s))(x)\,ds}$.

For initial conditions of the form and can thus show that for all one exists, so ${\displaystyle u_{0}\in L^{1}\cap L^{\infty }}$ ${\displaystyle p>3}$ ${\displaystyle C\geq 0}$ ${\displaystyle \delta >0}$

out

${\displaystyle \|u_{0}(\cdot )\|_{L^{1}}+\|u_{0}(\cdot )\|_{L^{\infty }}\leq \delta }$

follows that

${\displaystyle \|u(\cdot ,t)\|_{L^{\infty }}\leq C(1+t)^{-{\frac {1}{2}}}}$

${\displaystyle \|u(\cdot ,t)\|_{L^{1}}\leq C}$.

The condition is important because for smaller ones the reaction term is more important than the diffusion term and solutions of the equation would converge to infinity. ${\displaystyle p>3}$ ${\displaystyle p}$ ${\displaystyle f(u)=u^{p}}$ ${\displaystyle {\frac {\partial }{\partial x^{2}}}}$

For the consideration of general reaction diffusion equations, the reaction term and its behavior for solutions for large values ​​of compared to the behavior of the solutions of the linear diffusion equation play a major role. ${\displaystyle t}$

Individual evidence

1. ↑ Evans, Lawrence C .: Partial Differential equations . University Press, Hyderabad 2009, ISBN 978-0-8218-4859-3 .
2. ↑ ^{a b} Uecker, Hannes: Nonlinear PDEs: a dynamical systems approach . Providence, Rhode Island 2017, ISBN 978-1-4704-3613-1 .