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Session 17. Functional Analysis: relations to Complex Analysis and PDE |
Necessary and sufficient conditions for well-posedness of \(p\)-evolution equations |
| Chiara Boiti, University of Ferrara, Italy |
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For \(p\geq2\) we consider, in \([0,T]\times{\mathbb R}\), the \(p\)-evolution
operator \(P\) of the form
\begin{eqnarray*}
P(t,x,D_t,D_x)=D_t+a_p(t)D_x^p+\sum_{j=0}^{p-1}a_j(t,x)D_x^j,
\end{eqnarray*}
where \(a_p\in C([0,T];\mathbb R)\) and \(a_j\in C([0,T];{\mathcal B}^\infty)\)
for \(0\leq j\leq p-1\). We look for necessary and sufficient conditions for well-posedness in \(H^\infty\) of the associated Cauchy problem. The assumption that \(a_p\) is real valued means that the principal symbol, in the sense of Petrowski, has the real characteristic \(\tau=-a_p(t)\xi^p\) and is due to the Lax-Mizohata Theorem. Many results of well-posedness of the Cauchy problem are known when also the other coefficients \(a_j\), for \(0\leq j\leq p-1\), are real. When \(a_j\) are complex valued W. Ichinose proved, in the case \(p=2\), that some decay condition on \(\mathop{\rm Im}\nolimits a_{p-1}=\mathop{\rm Im}\nolimits a_1\) is necessary and sufficient for well-posedness in \(H^\infty\). In [1] we look for sufficient conditions for well-posedness in \(H^\infty\), obtaining a set of decay conditions, as \(x\to+\infty\), on \(\mathop{\rm Im}\nolimits D_x^\beta a_j\), for \(j\leq p-1\) and \([\beta/2]\leq j-1\). These results have been extended to the case of \(p\)-evolution equations of higher order in [2] and to semi-linear \(3\)-evolution equations in [3]. Then a necessary condition for well-posedness of the Cauchy problem
in \(H^\infty\)
has been proved in [4], generalizing to the case \(p\geq2\) the
necessary condition given by Ichinos |
References
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