On the Cauchy problem for \(p\)-evolution equations with variable coefficients: a necessary condition for Gevrey well-posedness (Q6592815)

In the present paper, the authors turn to the following non-degenerate Cauchy problem for Schrödinger-type equations of higher order: \N\[\NP u:= D_t u+ a_p(t)D_x^p u +\sum_{j=1}^p a_{p-j}(t,x)D_x^{p-j} u=0,\,\,\,u(0,x)=u_0(x). \tag{1}\N\]\NHere \(p \geq 2\) and \(a_p(t) \neq 0\) for \(t \in [0,T]\), \(T \in (0,\infty)\). For the regularity of coefficients the authors suppose \(a_p \in C([0,T],\mathbb{R})\) and for the complex-valued coefficients \(a_{p-j}\in C([0,T],B^\infty(\mathbb{R}))\) for \(j=1,\dots,p\).\N\NThe authors are interested in necessary conditions for Gevrey well-posedness in Gevrey spaces \(H^\infty_{\theta}(\mathbb{R})\). The Gevrey spaces are characterized by some decay behavior of the Fourier transform. Well-posedness means that a suitable energy inequality (with some kind of ``loss of Gevrey regularity'') is satisfied. The authors propose some hierarchy of conditions for \(a_{p-j},\,j=1,\dots,p-1\). These conditions read as follows with suitable positive constants \(A, C, R\), \(\sigma_{p-j}\in [0,1]\), \(\sigma_{p-1}\in [0,1)\) and all \(\beta \in \mathbb{N}\): \N\begin{gather*} \N\Im a_{p-j}(t,x)\geq A \langle x \rangle^{-\sigma_{p-j}}\text{ for }x >R \ (\text{or }x<-R),\quad t\in [0,T],\tag{2}\\\N\big|\partial_x^{\beta} a_{p-j}(t,x)\big|\leq C^{\beta+1}\beta! \langle x \rangle^{-\beta}.\tag{3}\N\end{gather*}\NThe authors prove that the condition \N\[\N\Sigma \leq \frac{1}{\theta},\,\,\Sigma:=\max_{j=1,\dots,p-1}\big\{(p-1)(1-\sigma_{p-j})-j+1\big\}\tag{4}\N\]\Nis necessary for well-posedness in \(H^\infty_{\theta}(\mathbb{R})\). This condition gives an interplay between \(\sigma_{p-j}\) and \(\theta\) to be necessary for \(H^\infty_{\theta}(\mathbb{R})\) well-posedness. The main idea of the proof is to define a sequence of localized energies of solutions around the bicharacteristic strip of the root of the principal symbol \(\xi^p\). From the assumed \(H^\infty_{\theta}(\mathbb{R})\) well-posedness of (1) one gets estimates of these energies to above. On the other hand by using (2) to (4) estimates to below of these localized energies are derived. Both estimates contradict each other.\N\NAfter this paper, a challenging problem is to study instead of (1) the Cauchy problem \N\[\ND_t u+ a_p(t,x)D_x^p u +\sum_{j=1}^p a_{p-j}(t,x)D_x^{p-j} u=0,\quad u(0,x)=u_0(x)\N\]\Nfrom the point of view of sufficient and necessary conditions for the \(H^\infty\) well-posedness or Gevrey well-posedness.