×

Ill-posedness for one-dimensional wave maps at the critical regularity. (English) Zbl 0966.35135

Consider the Cauchy problem \[ \square\varphi+ \varphi(\partial_\mu\varphi\cdot \partial^\mu\varphi)= 0,\tag{1} \]
\[ \varphi(0)= f\in S^{m-1},\quad \varphi_t(0)= 0, \] where \(\varphi: \mathbb{R}^{n+1}\to S^{m-1}\). Let \(n=1\) and \(m>2\) and let \(\varepsilon> 0\) and \(C>0\). Then there exists an initial data \(f\) with \(\|f- e_1\|_{C^\infty_0}\lesssim \varepsilon\), \(f- e_1\) supported on \((-c,c)\) such that the solution \(\varphi\) of (1) satisfies \[ \|\varphi(T)\|_{\dot H^{1/2}}\to{\i}\text{as }T\to\infty. \] In particular, \(\|\varphi(T)\|_{\dot B^{1/2,1}_2}\to \infty\) as \(T\to \infty\). In fact, the norms grow like \((\log T)^{1/2}\) and \(\log T\), respectively.

MSC:

35R25 Ill-posed problems for PDEs
35B40 Asymptotic behavior of solutions to PDEs
58J45 Hyperbolic equations on manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv Link