On the existence of an asymptotically large number of stable dissipative patterns in parabolic systems with small diffusion
The authors study the characteristics of the dynamics of dissipative patterns in the system with small diffusion \[ \frac{\partial u}{\partial t}=\nu D\frac{\partial^2 u}{\partial c^2} +Au+F(u),\quad \frac{\partial u}{\partial x}\Biggl|_{x=0}=\frac{\partial u}{\partial x}|_{x=\pi}=0. \] Here \(u\in \mathbb{R}^2, D\) and \(A\) are Hurwit z matrices, \(\nu>0\) is a parameter responsible for the proportional chance of the diffusion coefficients, \(F\in C^\infty(\mathbb{R}^2,\mathbb{R}^2).\) It is assumed that \(u=0 \) is the unique space-homogeneous equilibrium state, globally stable in the framework of the point model \(\frac{\partial u}{\partial t}=Au+F(u).\) The space of initial conditions (phase space) of the system is \(C([0, \pi];\mathbb{R}^2).\) It is shown that there exist many dissipative patterns, some of them are stable.
nonlinear system of reaction-diffusion type
scientific article; zbMATH DE number 1526521