Pseudodifferential equations in the spaces of generalized periodic functions (Q2761553)

The Bessel integral-differential operator of fractional order is constructed in the space of generalized \(2\pi\)-periodic functions. The Cauchy problem for the equation NEWLINE\[NEWLINE\beta(x)\partial U(t, x)/\partial t+ P(B_{\alpha_j}) U(t,x)= 0,\quad (t,x)\in (0,\infty)\times \mathbb{R}^nNEWLINE\]NEWLINE is considered. Here, \(P(B_\alpha)= \sum^m_{j=1} B_{\alpha_j}\), \(m\in\mathbb{N}\), \(\max_{1\leq j\leq m}\{\alpha_j\}> 0\); \(B_\alpha\) is a Bessel integral-differential operator of fractional order; \(\beta: (0,\infty)\to \mathbb{R}\) is a continuous positive function such that \(\int^t_0 \beta^{-1}(r) dr= O(t^k)\), \(k> 0\). Properties of solutions of the considered Cauchy problem are studied.