On self-similar patterns in coupled parabolic systems as non-equilibrium steady states (Q6543745)

The authors investigate reaction-diffusion systems and other related dissipative systems on unbounded domains. They discuss the self-similarity and asymptotical behavior. Self-similar behavior is a known phenomenon in extended systems. The solutions are usually considered with trivial behavior at infinity in the case of finite mass or energy of the considered physical system.\N\NThe authors of the present paper discuss the asymptotic self-similar behavior and show that it can occur in three different ways. Given three cases which can be distinguished after transforming into scaling variables: (A) The transformed system is autonomous and its steady state is a classical self-similar solution. (B) The transformed system converges to an autonomous system having suitable steady states. (C) An exponentially growing term generates a constraint that generates a constrained steady state. Constrained self-similar profiles occur mainly in systems of PDEs where diffusion and reaction terms scale differently.\N\NIt seems that the authors suppose that case (A) often occurs in scalar equations while cases (B) and (C) are more common in coupled systems of equations. Moreover, they consider the case of nonzero boundary conditions at infinity, which leads to systems with infinite mass displaying a richer structure than finite-mass systems.\N\NFor the evolution equation \((1)\) \(\tilde{u}_t=\tilde{f}(t,x, \tilde{u},\tilde{\nabla}\tilde{u}. \ldots ,\tilde{\nabla}^k\tilde{u})\) per definition, the solution \(\tilde{u}\) is called self-similar, in the sense of Barenblatt, if it can be written in the form \(\tilde{u}(t,x)= (1+t)^{-\alpha}U(x/(1+t)^{\beta})\) for a function \(U\) and scaling exponents \(\alpha \) and \(\beta \), which are suitably chosen, for instance, in order to guarantee mass conservation. The function \(U\) is called the profile of the self-similar solution. This concept is well-known, as it already finds an application for simple problems. To classify different types of self-similarity one has to transform the system into suitable scaling coordinates in which the self-similar profile appears as a steady pattern. The new coordinates \((\tau ,y)\) can be obtained by \(\tau = \log{(1+t)}\) and \(y=x/(1+t)^{\beta}\). After replacing \(u(\tau ,y)= (1+t)^{\alpha}\tilde{u}(t,x)\) in \((1)\) then obtain \((2)\) \(u_{\tau}=f(\tau ,y,u,\nabla u, \ldots ,\nabla^{k}u)\). Here \(\nabla\) concerns spatial derivatives w.r.t. \(y\). When the general structure of the transformed system has the form \(\boldsymbol{u}_{\tau}=\boldsymbol{f} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})+ e^{\gamma\tau} \boldsymbol{g} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})\), \(\gamma > 0\), and if \(\boldsymbol{u}(\tau ,y)\to \boldsymbol{u}(y)\) as \(\tau\to\infty\), then the constrained self-similar profile \(\boldsymbol{u}\) should satisfy \(\boldsymbol{g} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})=0\) and \(\boldsymbol{f} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})+ \boldsymbol{\lambda }(y)=0\), \(y\in\mathbb{R}^d\).\N\NNext the authors study systems on the unbounded real line that have the property that their restriction to a finite domain has a Lyapunov function and a gradient structure. Then the system reach local equilibrium on a rather fast time scale, but on unbounded domains with an infinite amount of mass or energy, it leads to a persistent mass or energy flow. It turns out that no true equilibrium is reached globally. In suitably rescaled variables, however, the solutions to the transformed system converge to non-equilibrium steady states that correspond to asymptotically self-similar behavior in the original system.