Fujita exponents on quantum Euclidean spaces

TL;DR

This paper extends the classical Fujita theorem to quantum Euclidean spaces, identifying critical exponents for the blow-up versus global existence of solutions to a nonlinear heat equation in a noncommutative setting.

Contribution

It introduces a noncommutative analogue of the Fujita theorem and establishes a key inequality in semifinite von Neumann algebras for analyzing nonlinear equations.

Findings

Identified the critical exponent for blow-up and global existence in quantum Euclidean spaces.

Proved a fundamental inequality in semifinite von Neumann algebras.

Demonstrated well-posedness results for nonlinear heat equations in noncommutative spaces.

Abstract

We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting.