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On the behavior of Floquet exponents of a kind of periodic evolution problems. (English) Zbl 0792.35026

The aim of this paper is to consider time periodic evolution PDE systems of the type \[ \begin{matrix} u_ t(t,x) = A(t,x,D) a(t) u(t,x) + b(t) u(t,x), & t > 0,\;x \in \Omega\\ B(x,D) a(t) u(t,x)\mid_{x\in \partial \Omega} = 0, & t > 0,\end{matrix}\tag{1} \] where \(\Omega\) is a bounded region in \(\mathbb{R}^ n\) with sufficiently smooth boundary, \(u(t,x)\) is a function on \(\mathbb{R} \times \overline{\Omega}\) with values in \(\mathbb{C}^ n\), \(a(t)\) and \(b(t)\) are periodic \(n \times n\)-matrix functions, differential operators \(A\) and \(B\) have scalar coefficients, and the scalar problem \[ \begin{aligned} v_ t & = Av\\ Bu\mid_{x \in \partial \Omega} & = 0\end{aligned} \] is in a sense hypoelliptic (for example, parabolic). The matrix \(a(t)\) is supposed to have real spectrum without adjoint vectors corresponding to zero eigenvalue. We assume that the period is equal to 1. The main assumption which implies some new features of (1) is that the determinant of \(a(t)\) is identically zero, and the kernel of \(a(t)\) has constant dimension.
A nonzero complex number \(\chi\) is said to be a Floquet exponent for a 1- periodic problem if this problem has a nonzero solution satisfying the condition \(u(t+1) = \chi u(t)\). In other words, there exists a nonzero solution of the type \(u(t) = g(t) \text{exp}(i\lambda t)\), where \(g(t)\) is 1-periodic and \(\text{exp}(i\lambda) = \chi\). The number \(\lambda\) is said to be quasimomentum. We prove that the set of Floquet exponents of (1) is either discrete with only possible accumulation points 0, \(\infty\), \(\chi_ 1,\dots,\chi_ m\) (where \(\dim\text{Ker }a(t) \equiv m\)) or contains \(\mathbb{C} \setminus \{0,\chi_ 1,\dots,\chi_ m\}\). Here \(\chi_ 1,\dots,\chi_ m\) are the Floquet exponents for an \(m\)- dimensional system of ordinary differential equations constructed by means of (1). If we have an exponential bound for solutions of (1) (for example, if (1) is parabolic), the set is discrete with the set of accumulation points \(\{0,\chi_ 1,\dots,\chi_ m\}\).

MSC:

35G10 Initial value problems for linear higher-order PDEs
65H10 Numerical computation of solutions to systems of equations
34G10 Linear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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