The Fourier method in the Cauchy problem and absolutely representing systems of exponentials. I (Q2703857)

The author applies the Fourier method to solve the following Cauchy problem for the following differential equations with variable coefficients NEWLINE\[NEWLINE P\left(\frac{\partial}{\partial z_{2}}\right)\frac{\partial u}{\partial z_{1}} = \sum_{k=0}^m a_{k} (z_{1}) \frac{\partial ^k u}{{\partial z_{2} }^k}, \quad m\geq 1, NEWLINE\]NEWLINE where \(P=\sum_{l=0}^p b_{l}z^l\) is a polynomial of degree \(p\geq 0\), with the initial condition \(u(0,z_2)= \varphi(z_2)\). This extends his previous results on the Cauchy problems for the above differential equations with constant coefficients. Precisely speaking, if \(E\) is a complete separable locally convex space of complex-valued functions defined on subsets of \(\mathbb R\) or \(\mathbb C\), having the several properties and \(H\) is a space of analytic functions with a suitable condition; he showed the following result: The solvability of the Cauchy problem depends on the determining parameter \(s := m - p\) as follows. NEWLINENEWLINENEWLINEIf \(s\leq 0,\) then the Cauchy problem is solvable in some space \(\{H;E\}\) for any initial function \(\varphi\) in \(E\). NEWLINENEWLINENEWLINEIf \(s=1\), then we need one more condition on \(E\) (Theorem 2.1) or some conditions relating \(E\) to \(H\) (Theorems 2.2 and 2.3). So the class of ``admissible'' space \(E\) is narrower. NEWLINENEWLINENEWLINEIf \(s\geq 2,\) then the class of spaces \(E\) of initial functions for which the Cauchy problem has a solution analytic with respect to the ``initial'' variable \(z_1\) is quite narrow and is subspace of the space \([\frac{s}{s-1},\infty)\).