Dynamics of stationary structures in the canonical parabolic problem

This work researches the dynamics of stationary structures of equation of reaction-diffusion type with the diffusion coefficient $\mu \to 0$ in the interval $[0, \pi]$ and Neumann conditions. The stationary solution $\phi_\infty = 0$ of this equation is unstable, and $\phi_0 = 1$, $\widehat{\phi_0} = −1$ – are stable. When $(k + 1) − 2 < \mu < k−2, k = 1, 2, ... ,$ many stationary solutions contain exactly $k$-pairs of spatially inhomogeneous solutions $\phi_i, \widehat{\phi_i} = −\phi_i, 1 \le i \le k$. The dimension of the unstable manifolds $\phi_i$, $\widehat{\phi_i}$ is equal to $i$. The function $\phi_i$ approaches, as $\mu \to 0$, the step function with the values of $1, −1$ and the breakpoints – zeros $\phi_i$.
By method of center manifold the theorem on the existence and stability of spatially inhomogeneous stationary solutions $\phi_1(x, \mu), \phi_1(\pi−x, \mu),$ branching off from the zero solution with $\mu = 1$ has been proved. The analysis conducted has shown that the asymptotic expansion of the solution $\phi_1(x, \mu)$ in the vicinity of $\mu = 1$ obtained in the theorem is an approximate solution of the problem under consideration with the sufficiently wide range of the parameter $\mu$. The theorem assertions on the existence, shape and stability are of local as for the parameter $\mu$ nature. We should note that with sufficiently small $\mu$ function $\phi_1(x, \mu)$ is similar to a step function with values $1, −1$ and one transition point $\frac{\pi}{2}$. The solutions $\phi_k(x, \mu), k = 2, 3, ... ,$ are built being based on $\phi_1(x, \mu)$, using the principle of similarity. In particular, the solution $\phi_2(x, \mu)$ has been built. We should emphasize that with small $\mu\>\phi_2(x, \mu)$ is the solution of the type of internal transition layer with two transition points $\left(\frac{\pi}{4},\frac{3\pi}{4}\right).$ Using “Mathematica” package, the original solution of the parabolic problem has been built, where the approximate representation $\phi_1(x, \mu)$ with $\mu = 0.01$ is taken as a primary function. The solution obtained retains its shape during quite a long time equal to $10^{19}$. Also by means of “Mathematica” package the solution of the problem is built, where the function found by means of the principle $\phi_2(x, \mu)$ is used with $\mu = 0.01.$ The solution obtained is not changed for quite a long time $\approx 10^{27}$. Thus, these solutions generate metastable structures (slowly varying solutions).
The questions about existence of metastable structures of the initial problem are considered by the authors J. Carr and R.L. Pego, G. Fusco and J.K. Hale.
In this study it has been found that for solution of the problem of constructing the canonical structures of stationary parabolic problem with small \mu application of the center manifold method leads to qualitatively and quantitatively correct results.

Keywords: parabolic problem, the method of central manifolds, instability, internal shock layer, stationary solutions

UDC: 
517.957