Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables

TL;DR

This paper investigates p-adic nonlinear evolution equations, including the p-adic porous medium equation, establishing existence and uniqueness of solutions using semigroup and m-accretivity methods.

Contribution

It introduces a novel approach to solving p-adic nonlinear evolution equations by constructing a linear Markov semigroup and proving m-accretivity for the nonlinear operator.

Findings

Established existence and uniqueness of mild solutions for p-adic porous medium equations.

Constructed a linear Markov semigroup on a p-adic ball.

Proved m-accretivity of the nonlinear operator.

Abstract

We study the Cauchy problem for $p$-adic nonlinear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the p-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a p-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result is equivalent to the existence and uniqueness of a mild solution of the Cauchy problem of a nonlinear equation of the porous medium type.