# A simple necessary and sufficient condition for well-posedness of $2\times 2$ differential systems with time-dependent coefficients

• Volume: 9-B, Issue: 1, page 215-220
• ISSN: 0392-4041

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## Abstract

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Given the Cauchy Problem $\partial_{t}u(x,t)+A(t)\partial_{x}u(x,t)=0\qquad u(0,x)=u_{0}(x)$ Nishitani [N], by making use of a change of basis by a constant matrix, transformed the real, analytic, hyperbolic matrix $A(t)=\left(\begin{array}[]{cc}d(t)&a(t)\\ b(t)&-d(t)\\ \end{array}\right)\qquad t\in[0,T]$ into the complex matrix $A^{\sharp}(t)=\left(\begin{array}[]{cc}c^{\sharp}(t)&a^{\sharp}(t)\\ a^{\sharp}(t)&-c^{\sharp}(t)\\ \end{array}\right)=\left(\begin{array}[]{cc}i\frac{a-b}{2}&\frac{a+b}{2}+id\\ \frac{a+b}{2}-id&-i\frac{a-b}{2}\\ \end{array}\right)$and showed that the given Cauchy Problem is well posed in $C^{\infty}$ in a neighborhood ofzero if and only if (see also [MS]) the following condition $h|a^{\sharp}|^{2}\geq Ct^{2}|D^{\sharp}|^{2}$ is satisfied, where $D^{\sharp}=\dot{a}^{\sharp}c^{\sharp}-\dot{c}^{\sharp}a^{\sharp}\ \text{e}h=-% detA=|a^{\sharp}|^{2}-|c^{\sharp}|^{2}.$ In this short note, we give a very simple condition, which is equivalent to that of Nishitani (and then a necessary and sufficient for the Well-Posedness), but where only the elements of appear and not their derivatives.

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