Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions
The author proves the existence of an inertial manifold for systems of the form: \[ u_t-d\Delta u+f(x,u)+g(x,v)=0,\;x\in\Omega\subseteq R^N,\quad v_t+\sigma(x)v+h(x,u)=0\tag{1} \] with \(\Omega\) a smooth bounded domain, and the nonlinearities \(f,g,h\) subject to a variety of conditions. The Fitz-Hugh-Nagumo equations are a special case of (1). The theorem which asserts the existence of an inertial manifold for (1) is on page 40 of this long and technically involved paper. In fact, the first half of the paper is devoted to the construction of inertial manifolds for an abstract system which reads as follows: \[ p'=F(p,q,v),\quad q'=-Aq+G(p,q,v),\quad v'=-B(p,q)v+H(p,q).\tag{2} \] The system is based on two Hilbert spaces \(H_1,H_2\) with \(H_1\) the orthogonal sum of two closed subspaces \(E,L\), i.e. \(H_1=E\oplus L\). One assumes \(p\in E\), \(q\in L\), \(v\in H_2\). The nonlinearities \(F,G,H\) are uniformly bounded and Lipschitz continuous, while \(B(p,q)\) is a bounded linear operator on \(H_2\), depending Lipschitz continuously on \(p,q\) and such that \(\gamma\| v\|^2\leq\langle v,B(p,q)v\rangle\leq\Gamma\| v\|^2\), \(p\in E\), \(q\in L\), \(v\in H_2\) for some \(\gamma,\Gamma\). Finally, \(A\) is selfadjoint, positive on \(L\). Under these assumptions a global existence and uniqueness theorem for solutions of (2) holds. The author now imposes five conditions on the data of (2), too long to be stated here. Based on these conditions, he proves the existence of a global inertial manifold for (2). The proof is quite involved due to the fact that the manifold has infinite dimensions. Based on some considerations on global attractors, a standard smoothing procedure utlimately leads to the existence of inertial manifolds of (1). The paper concludes with some examples.
scientific article; zbMATH DE number 1206644
scientific article