Well posedness in \(C^ \infty\) for a weakly hyperbolic second order equation
This paper, which represents an interesting contribution to the theory of weakly hyperbolic linear equations, is devoted to the one-dimensional Cauchy problem \[ u_{tt} = \bigl( a(t,x) u_ x \bigr)_ x, \quad u(0,x) = u_ 0(x), \quad u_ t(0,x) = u_ 1(x), \tag{*} \] where \(a(t,x)\) is a \({\mathcal C}^ \infty\) function satisfying \(0 \leq a (t,x) \leq \Lambda_ 0\) on the strip \(S_ T = [0,T] \times \mathbb{R}\). It is known [\textit{F. Colombini} and \textit{S. Spagnolo}, Acta Math. 148, 243-253 (1982; Zbl 0517.35053)] that there exists a smooth coefficient \(a(t,x) \equiv a(t)\) for which Problem \((*)\) is not well posed in \({\mathcal C}^ \infty (\mathbb{R})\), even locally in time. On the other hand, a result of O. A. Oleinik [Commun. Pure Appl. Math. 23, 569-586 (1970; Zbl 0193.386)] ensures in particular that \((*)\) is globally well-posed in \({\mathcal C}^ \infty\) when there exist \(0 = t_ 0 < t_ 1 < \cdots < t_ \nu = T\) such that, in each strip \([t_{i-1}, t_ i] \times \mathbb{R}\), the alternative \(a_ t(t,x) \geq - Ka (t,x)\) or \(a_ t(t,x) \leq Ka (t,x)\) holds for some constant \(K > 0\); in particular when \(a(t,x) \equiv a(x)\), or when \(a(t,x) \equiv a(t)\) with \(a(t)\) analytic on \([0,T]\). In the present paper, the author improves Oleinik's result, by showing that \((*)\) is globally well-posed in \({\mathcal C}^ \infty\) whenever, for all \(r>0\), the rectangle \([0,T] \times [- r,r]\) can be split in a finite number of normal domains \(G_{r,i} = \{(t,x) : | x | \leq r\), \(\varphi_{i-1} (x) \leq t \leq \varphi_ i (x)\}\), \(0 \equiv \varphi_ 0 (x) \leq \cdots \leq \varphi_{i-1} (x) \leq \varphi_ i (x) \leq \cdots \leq \varphi_ \nu (x) \equiv T\), whose limiting curves are time-like with respect to the equation (i.e. \(a (\varphi_ i (x),x) \cdot \varphi_ i' (x)^ 2 \leq 1)\), in such a way that the coefficient \(a(t,x)\) satisfies the above alternative in each one of the \(G_{r,i}\)'s. The proof of this result is based on a delicate interplay between the classical energy methods and the Oleinik's technique. As a corollary, the author obtains the global well-posedness in \({\mathcal C}^ \infty\) whenever \(a(t,x)\) is analytic on \(S_ T\), thus extending a previous result of \textit{T. Nishitani} [Commun. Partial Differ. Equations 5, 1273-1296 (1980; Zbl 0497.35053)] who had proved the local solvability in time. The extension of this result to several space dimensions is still an open question.
global well-posedness in \({\mathcal C}^ \infty\)
scientific article; zbMATH DE number 627425