Cauchy problem for a parabolic pseudodifferential higher-order equation with pulse action

Abstract

UDC 517.956 We consider the Cauchy problem and the problem with pulse action for a pseudodifferential higherorder equation with respect to t . We construct solutions of these problems, study their properties, and prove a theorem on correctness. The theory of pseudodifferential operators and pseudodifferential equations, whose modern form was formed in the middle of the 1960s, is studied in many works [2–8, 11, 12]. After determination that pseudodifferential operators are closely connected with problems of analysis and contemporary mathematical physics, especially in the theory of elliptic boundary-value problems [10], the number of such works has considerably increased. Linear parabolic pseudodifferential equations with nonsmooth symbols were defined by Eidel’man, Drin’, and Iwasaki at the beginning of the 1970s in [2, 3, 8, 9, 11, 12]. The symbols of these pseudodifferential operators are nonsmooth at the point σ= 0 , σ∈ R n . For this reason, standard methods, which are used for pseudodifferential operators with smooth symbols, cannot be applied to the investigation of problems for these pseudodifferential equations. Investigation of these equations with constant (independent of the space coordinates x ∈R n and the time coordinate t ∈(0,T ]) homogeneous symbol was originated in [8]. The fundamental solution of the Cauchy problem for these equations was determined with the use of the Fourier transformation. In [7], Fedoryuk establishes that the exact asymptotics of the fundamental solution of the Cauchy problem as x →∞ is power rather than exponential as for differential equations. In [3], Schauder estimates are obtained and the correct solvability of the Cauchy problem in classes of Holder functions is established. For the subsequent development of this theory, Kochubei’s works [4, 11] were very important. In these works, for the first time, it is noted that pseudodifferential operators with nonsmooth symbols can be interpreted as hypersingular integral operations. This enabled one to use the well-developed theory of hypersingular integral operations for the investigation of the Cauchy problem. Kochubei constructed and studied fundamental solutions of the Cauchy problem, proved theorems on solvability of the Cauchy problem in classes of functions with power increase as x →∞ , and indicated connections of the obtained results with theory of random processes.