Operators of \(p\)-evolution with nonregular coefficients in the time variable
The authors study \(p\)-evolution operators \[ P= D^m_t+ \sum^{m-1}_{j=0} A_j(t, x,D_x)D^j_t \] in two different cases \(\bullet\) \(A_j\in C([0, T]\), \(S^{p(m-j)}(\mathbb{R}^n))\) -- well-posedness in Sobolev spaces, \(\bullet\) \(A_j\in C([0, T]; S^{p(m-j)}_{\sigma_0}(\mathbb{R}^n))\)-well-posedness in Gevrey spaces, where \(\sigma_0\geq 1\) denotes the order of Gevrey spaces with respect to the spatial variables. To characterize their class of \(p\)-evolution operators the authors suppose in both cases the existence of a pseudodifferential operator \(P^0(t,x,D_t,D_x)\) whose symbol \(P^0(t,x, \tau,\xi)\) satisfies: \(\bullet\) \(P^0(t,x,\tau,\xi)= \prod^m_{j=1} (\tau-\lambda_j(t, x,\xi))\), \(\bullet\) \(\lambda_j(t, x,\xi)= \lambda^0_j(t, \xi)+ \lambda^1_j(t,x,\xi)\), where \(\lambda^0_j\in \text{Log\,Lip}([0, T]; S^p(\mathbb{R}^n))\), \(\lambda^1_j\in \text{Log\,}C^{1/p}([0, T]\); \(S^1(\mathbb{R}^n))\) if one is interested in well-posedness in Sobolev spaces: \(\lambda^0_j\in C^\kappa([0,T]; S^p(\mathbb{R}^n))\), \(\lambda^1_j\in C^{\kappa+ 1/p-1}([0, T]\); \(S^1_{\sigma_0}(\mathbb{R}^n))\), \(\kappa> 1-1/p\), if one is interested in well-posedness in Gevrey spaces, \(\bullet\) \(\lambda_j\) are real if \(p> 2\), if \(p\leq 2\), then some conditions for \(\text{Im\,}\lambda_j\) allow to apply the Fefferman-Phong inequality, \(\bullet\) \(|\lambda^0_j(t, \xi)- \lambda^0_k(t,\xi)|\geq C|\xi|^p\), \(c> 0\), \(j\neq k\), \(|\xi|\) large, \(\bullet\) \(P-P^0= \sum^{m-1}_{j=0} A^1_j(t, x,D_x) D^j_t\), where \(A^1_j\in C([0, T]\), \(S^{p(m-1-j)}(\mathbb{R}^n))\) in the Sobolev case and \(A^1_j\in C([0, T]\), \(S^{p(m-1-j)+ q_0}_{\sigma_0}(\mathbb{R}^n))\), \(q_0< 1\), in the Gevrey case. If given \(p\)-evolution operators satisfy these conditions, then well-posedness results in scales of Sobolev spaces or Gevrey spaces can be proved, where a loss of regularity appears. The approach to prove these well-posedness results is based on a diagonalization technique coupled with regularizations of \(\lambda_j\) to guarantee \(C^1([0, T];\dots)\) behaviour. Suitable transformations containing the time-dependent loss of regularity transfer the starting Cauchy problem to a pseudodifferential evolution system with a nonnegative matrix-valued operator. After an application of the sharp Gårding's inequality Gronwall's inequality gives the desired well-posedness result.
scientific article; zbMATH DE number 2092987