Properties of integrals which have the type of derivatives of volume potentials for parabolic systems with degeneration on the initial hyperplane (Q1605551)

The authors study properties of integrals of the type \[ u(t,x)=\int_0^t\frac{d\tau}{\alpha(\tau)}\int_{{\mathbb R}^n} M(t,\tau,x-\xi) f(\tau,\xi) d\xi,\quad t\in(0,T],\;x\in\mathbb R^n, \] where \(\alpha:[0,T]\to[0,\infty)\) and \(\beta:[0,T]\to[0,\infty)\) are continuous functions such that \(\alpha(0)\beta(0)=0\) and \(\alpha(t)>0,\beta(t)>0\) for \(t>0\). The kernel \(M\) has properties of derivatives of the fundamental matrix for solutions \(Z\) of the Cauchy problem for parabolic by Petrovsky systems of \(N\) equations with degeneration on the initial hyperplane of the form \[ \alpha(t)\partial_tu(t,x)=\beta(t)\sum_{1\leq|k|\leq 2b} a_k(t)\partial_x^ku(t,x)+a_0u(t,x)+f(t,x),\;t\in(0,T],\;x\in\mathbb R^n. \] It is proved that the function \(u(t,x)\) belongs to certain functional space when the function \(f(t,x)\) belongs to the appropriate space.