P-Adic semilinear evolutionary pseudodifferential equations in Lizorkin spaces

Abstract

There are many applications of p -adic analysis in quantum mechanics, string theory, stochastics, the theory of dynamical systems, and cognitive sciences (3-7, 9, 13-15). The systematic use of p -adic analysis in applications began with pioneering works of V.S. Vladimirov and I.V. Volovich. In p -adic analysis, which is related to the mappingp → � , differentia- tion is not defined; for this reason, p -adic models extensively use pseudodifferential operators, in par- ticular, the fractional operator D α . Up to now, in such models, only linear pseudodifferential equations have been considered. In this paper, we develop a new approach to constructing solutions to a large class of p -adic semilinear evolutionary pseudodifferential equations, which is based on the theory of p -adic pseudodifferential operators in the Lizorkin space (1, 2) and on the theory of p -adic wavelets (8) (see (1) for a multidimensional analogue). We use the notation and results of classical book (13).