Global Existence for a Nonlinear System with Fractional Laplacian in Banach Space

TL;DR

This paper establishes global existence results for a class of nonlinear fractional dissipative equations and systems in Banach spaces, based on scaling properties and multilinear forms.

Contribution

It extends existing methods to prove global solutions for fractional Laplacian systems with multilinear nonlinearities in Banach spaces.

Findings

Proves global existence under specific scaling conditions.

Extends results from single equations to coupled systems.

Provides a framework for analyzing fractional PDEs in Banach spaces.

Abstract

We consider the cauchy problem for the fractional power dissipative equation $u_t+(-\Delta )^{\beta/2} u=F(u)$, where $\beta>0$ and $F(u)=B(u, ...,u)$ and $B$ is a multilinear form on a Banach space $E$. We show a global existence result assuming some properties of scaling degree of the multilinear form and the norm of the space $E$. We extend the ideas used for the treating of the equation to determine the global existence for the system $u_t+(-\Delta)^{\beta/2}= F(v)$, $v_t+(-\Delta )^{\beta/2}v=G(u)$ where $F(u)=B_1(u,...,u), G(v)=B_2(v,...,v)$