Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0820.35097
D'Ancona, Piero
Well posedness in $C\sp \infty$ for a weakly hyperbolic second order equation. (English)
[J] Rend. Semin. Mat. Univ. Padova 91, 65-83 (1994). ISSN 0041-8994

This paper, which represents an interesting contribution to the theory of weakly hyperbolic linear equations, is devoted to the one-dimensional Cauchy problem $$u\sb{tt} = \bigl( a(t,x) u\sb x \bigr)\sb x, \quad u(0,x) = u\sb 0(x), \quad u\sb t(0,x) = u\sb 1(x), \tag *$$ where $a(t,x)$ is a ${\cal C}\sp \infty$ function satisfying $0 \le a (t,x) \le \Lambda\sb 0$ on the strip $S\sb T = [0,T] \times \bbfR$. It is known [{\it F. Colombini} and {\it 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 ${\cal C}\sp \infty (\bbfR)$, 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 ${\cal C}\sp \infty$ when there exist $0 = t\sb 0 < t\sb 1 < \cdots < t\sb \nu = T$ such that, in each strip $[t\sb{i-1}, t\sb i] \times \bbfR$, the alternative $a\sb t(t,x) \ge - Ka (t,x)$ or $a\sb t(t,x) \le Ka (t,x)$ 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 $[0,T]$.\par In the present paper, the author improves Oleinik's result, by showing that $(*)$ is globally well-posed in ${\cal C}\sp \infty$ whenever, for all $r>0$, the rectangle $[0,T] \times [- r,r]$ can be split in a finite number of normal domains $G\sb{r,i} = \{(t,x) : \vert x \vert \le r$, $\varphi\sb{i-1} (x) \le t \le \varphi\sb i (x)\}$, $0 \equiv \varphi\sb 0 (x) \le \cdots \le \varphi\sb{i-1} (x) \le \varphi\sb i (x) \le \cdots \le \varphi\sb \nu (x) \equiv T$, whose limiting curves are time-like with respect to the equation (i.e. $a (\varphi\sb i (x),x) \cdot \varphi\sb i' (x)\sp 2 \le 1)$, in such a way that the coefficient $a(t,x)$ satisfies the above alternative in each one of the $G\sb{r,i}$'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 ${\cal C}\sp \infty$ whenever $a(t,x)$ is analytic on $S\sb T$, thus extending a previous result of {\it 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.
[S.A.Spagnolo (Pisa)]

MSC 2000:: *35L80 Hyperbolic equations of degenerate type
35L15 Second order hyperbolic equations, initial value problems

Keywords: Oleinik conditions; global well-posedness in ${\cal C}\sp \infty$

Citations: Zbl 0517.35053; Zbl 0193.386; Zbl 0497.35053

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]