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Stability and diffusive dynamics on extended domains. (English) Zbl 1004.35018
Fiedler, Bernold (ed.), Ergodic theory, analysis, and efficient simulation of dynamical systems. Berlin: Springer. 563-583 (2001).
This is an interesting survey article on the stability behaviour of a class of nonlinear diffusive equations on an unbounded domain \(\Omega\); here \(\Omega=\mathbb{R}^n\), \(n=1,2\) in most cases. The authors first consider the heat equation \(\partial_tu= \partial^2_xu\) on \(\mathbb{R}\) and the related \(L^p-L^q\) estimates according to which \[ \|u(t)\|_{L^p}\leq Ct^{-1/2r}\bigl\|u(0)\bigr \|_{L^q},\tag{1} \] for \(p>q\), \(p^{-1}=q^{-1}-r^{-1}\). The Sobolev spaces \(H^m(n)\) are defined via: \(u\in H^m(n)\) iff \(u\in H^m\) and \(\|\rho^n u\|_{H^m}< \infty\), where \(\rho(x)= (1+x^2)^{1/2}\); they are needed in order to formulate the later asymptotic results correctly. Based on these preparatory remarks, the equation \[ \partial_tu= \partial^2_xu+ cu^p,\;u(0)=u_0,\;c\neq 0\tag{2} \] can be handled for \(p>3\) by any of three methods: \(L^\infty-L^1\) estimates, Lyapunov functions, renormalisation processes. It is indicated that the condition \(p>3\) implies that the nonlinear term \(cu^p\) has negligible influence on the evolution, what justifies to call \(p\) the degree of irrelevance. In a subsequent section it is sketched how these methods extend to the Ginzburg-Landau equation: \[ \partial_tu= \partial^2_xu+u-|u|^2u,\;u(t,x)\in\mathbb{C},\;u(0,x)=u_0(x). \tag{3} \] Equation (3) has a family of equilibrium solutions (rolls) \(u_{q\beta}= (1-q^2)e^{i(q x+\beta)}\), \(q\in (-1,+1)\). It is sketched how the methods used for (2) can be used to discuss the stability of rolls. The next section, which is devoted to pattern forming systems, starts with a brief outline of Bloch space theory. The Swift-Hohenberg equation is then considered: \[ \partial_tu= -(1+\Delta)^2u +\varepsilon^2u-u^3,\;t\geq 0,\;x\in\mathbb{R}^d,\;u\in\mathbb{R} \tag{4} \] with \(\varepsilon>0\) a small parameter. If \(d=1\), (4) has a family \(u_{\varepsilon\chi}\), \(\chi\in(-\varepsilon, +\varepsilon)\) of equilibrium solutions. It is then outlined how to discuss the stability of \(u_{\varepsilon \chi}\) in terms of Bloch space theory. A theorem, due to one of the authors, is then given which describes the asymptotic behaviour of solutions \(u(x,t)\) of (4) with \(u(x,0)\) close to \(u_{\varepsilon \chi}\) in terms of the space \(H^2(2)\). The section closes with a review of the case \(d=2\), which is again analysed in terms of Bloch space theory. The paper ends with some remarks on the 2D-Bénard problem and with a review of open problems.
This article is an interesting and stimulating review of some recent results in the stability theory of diffusive processes on unbounded domains. The presentation is concise and the interested reader is advised to consult some of the many references.
For the entire collection see [Zbl 0968.00013].

35B35 Stability in context of PDEs
35K55 Nonlinear parabolic equations
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q35 PDEs in connection with fluid mechanics
35K15 Initial value problems for second-order parabolic equations