Cauchy problem for heat-transfer equation with irregular elliptic operator (Q912284)

The author continues the study of the abstract Cauchy problem \[ \partial u(x,t)/\partial t=J(u''_ x(x,t)),\quad u(x,0)=\lim_{t\to 0}u(x,t)=\phi (x), \] where (x,t)\(\in H\times (0,\infty)\), H a separable Hilbert space, \(\phi\in {\mathcal N}\), \({\mathcal N}^ a \)class of functions defined on H, with some special properties, J is a positive linear functional defined on \(B_ c(H)\); \(B_ c(H)\) is the Banach space of linear selfadjoint bounded operators defined on H. The operator J is not regular in a certain sense. An expression of the solution u(x,t) is given (Theorem 1) and also a uniqueness theorem (Theorem 3).