Monotonicity of stable solutions in shadow systems (Q2750935)

The authors study the following reaction-diffusion system NEWLINE\[NEWLINE\begin{aligned} & u_t= u_{xx}+ f(u, v,t),\quad x\in (0,1),\quad t> 0,\quad u_x(0, t)= u_x(1,t)= 0,\\ & v_t= \int^1_c g(u,v,t) dx,\end{aligned}\tag{1}NEWLINE\]NEWLINE where \(u(x,t)\in\mathbb{R}\) and \(v= v(t)\in \mathbb{R}^n\). This system appears as the limit in some sense of the reaction-diffusion system NEWLINE\[NEWLINEu_t= u_{xx}+ f(u, v,t),\quad v_t= \varepsilon^{-2} v_{xx}+ g(u,v,t)NEWLINE\]NEWLINE subject to homogeneous Neumann boundary conditions, as is shown in the cited references. The functions \(f\), \(g\) are subject to certain smoothness and boundedness conditions, labeled as (H1), (H2). The authors now study the stability of bounded solutions of (1). Since (1) is nonautonomous, this notion needs some care. Following \textit{D. Henry} [Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag (1981; Zbl 0456.35001)], they consider the evolution operators \(T(t, s)\), \(t> s\geq 0\) associated with the linearization of (1) at a given bounded solution \((u(t), v(t))\), \(t\geq 0\). This solution is then linearly stable if there are \(\lambda\), \(C>0\) such that \(\|T(t,s)\|\leq Ce^{-\lambda(t- s)}\), \(t> s\geq 0\). There is also a related notion of exponential instability which is given in the paper.NEWLINENEWLINENEWLINETwo conditions are imposed on the bounded conditions under scrutinyNEWLINENEWLINENEWLINE(A1) \((u(x,t), v(t))\) is uniformly bounded on \([0,1]\times [0,\infty)\),NEWLINENEWLINENEWLINE(A2) \(\|u_x(\varepsilon, t)\|_{L^\infty}\geq d\), \(t>0\), for some \(d>0\).NEWLINENEWLINENEWLINEThe main result of the paper states Theorem 1: Assume (H1), (H2) and let \((u(x,t), v(t))\), \(t\geq 0\) be a solution to (1) subject to (A1) and (A2). Then there is \(t_0\) such that \(u_x(x,t)\neq 0\) for all \((x,t)\in (0,1)\times (t_0,\infty)\).NEWLINENEWLINENEWLINEA corollary states that if \(f(u,v,t)\), \(g(u,v,t)\) are \(\tau\)-periodic for some \(\tau>0\) and sufficiently smooth, then any \(\tau\)-periodic solution of (1) is linearly exponentially unstable if it is specially inhomogeneous and nonmonotone. A number of further results are obtained. The technique of proof is based on a subtle analysis of the linearizations which occur in connection with (1) and related equations.