MULTIPOINT BY TIME PROBLEM FOR A CLASS OF EVOLUTION EQUATIONS IN S TYPE SPACE

Abstract

The goal of this paper is to study evolution equations of the parabolic type with operators $\displaystyle \varphi\Big(i \frac{\partial}{\partial x}\Big)$ built according to certain functions (different from polynomials), in particular, with operators of fractional differentiation. It is found that the restriction of such operators to certain $S$-type spaces match with pseudo-differential operators in such spaces constructed by these functions, which are multipliers in spaces that are Fourier transforms of $S$-type spaces. The well-posedness of the nonlocal multipoint by time problem is proved for such equations with initial functions that are elements of spaces of generalized functions of $S$-type. The properties of the fundamental solutions of the specified problem, the behavior of the solution at $t\to +\infty$ in spaces of $S'$-type (weak stabilization) were studied. We found conditions under which the solution stabilizes to zero uniformly on $\mathbb{R}$.