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JUN-PING SHI EXPOSITION OF ONE-DIMENSIONAL NONLINEAR DIFFUSIONS

In a very nice expository chapter of a book Jun-Ping Shi introduces in an elementary manner the stability issues of the nonlinear diffusion-reaction equation $\frac{\partial u}{\partial t} - D \frac{\partial^2}{\partial x^2} u - g(u) =0$ with $u(t,0)=u(t,L)=0$ and $u(0,x)=f(x)$ where $g(\cdot)$ is a nonlinear reaction term with an analysis of the ordinary differential equation counterpart $u'' - g(u) =0$.  A major issue is that it is the zeros of the function $g(u)=0$ that provide equilibrium solutions for example for $g(u) = \lambda u(1-u)$ then the equilibrium solutions when $t\rightarrow \infty$ are $u=0$ and $u=1$, the former being an unstable solution and the latter the stable one and the stability depends on $\lambda$ as a function of $L$, the length of the space interval.  The transition from the ordinary differential equation case to the partial differential system introduces some complexity where his Theorem 4.1 provides a criterion for stability in terms of a linear system:

So in the NONFRACTIONAL reaction-diffusion equation, there are classical methods to determine stability of solutions in terms of signs of eigenvalues of a system.  In terms of theory, this provides some some tools for dealing with nonlinearities more general than the Fisher-KPP type $g(u) = u(1-u)$ which I was missing for several months focused on the issues of models in continuous-time finance where the unrealistic case of KPP nonlinearity would tell us clearly that zero would be an unstable solution but we do not have clear nonlinearities empirically determined yet.