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Fujita exponents on quantum Euclidean spaces

Edward McDonald, Michael Ruzhansky, Serikbol Shaimardan, Kanat Tulenov

TL;DR

This work extends the classical Fujita theory for nonlinear heat equations to noncommutative geometry by studying the semilinear heat equation on quantum Euclidean spaces $L^{\infty}(\mathbb{R}^{d}_{\theta})$. It develops the noncommutative analytic framework (NC $L^{p}$ spaces, trace, differential calculus) and analyzes the NC heat semigroup via a Gaussian operator, establishing a Tauberian-type norm equivalence for the $L^{1}$-norm. A central contribution is a fundamental nonlinear inequality for $p$-th powers in semifinite von Neumann algebras, enabling global existence and local well-posedness theory through double operator integrals. The paper identifies the critical exponent $p_F=1+\frac{2}{d}$, proving blow-up for $p\le p_F$ and global existence for small data when $p>p_F$, thereby generalizing Fujita-type results to noncommutative spaces and providing tools for nonlinear PDEs in operator-algebra contexts.

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.

Fujita exponents on quantum Euclidean spaces

TL;DR

This work extends the classical Fujita theory for nonlinear heat equations to noncommutative geometry by studying the semilinear heat equation on quantum Euclidean spaces . It develops the noncommutative analytic framework (NC spaces, trace, differential calculus) and analyzes the NC heat semigroup via a Gaussian operator, establishing a Tauberian-type norm equivalence for the -norm. A central contribution is a fundamental nonlinear inequality for -th powers in semifinite von Neumann algebras, enabling global existence and local well-posedness theory through double operator integrals. The paper identifies the critical exponent , proving blow-up for and global existence for small data when , thereby generalizing Fujita-type results to noncommutative spaces and providing tools for nonlinear PDEs in operator-algebra contexts.

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.
Paper Structure (15 sections, 21 theorems, 206 equations)

This paper contains 15 sections, 21 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Noncommutative (NC) Euclidean space $L^{\infty}(\mathbb{R}^{d}_{\theta})$
  4. Noncommutative integration
  5. Trace on $L^{\infty}(\mathbb{R}_{\theta}^d)$
  6. Noncommutative $L^{p}(\mathbb{R}^{d}_{\theta})$ spaces
  7. Differential calculus on $L^{\infty}(\mathbb{R}^{d}_{\theta})$
  8. Heat kernel on quantum Euclidean spaces
  9. The Gaussian operator
  10. Equivalent form of $L^1$-norm
  11. Nonlinearity estimate
  12. Local existence of the solution
  13. Auxiliary results
  14. Fujita-type results
  15. Operator convexity

Key Result

Lemma 3.2

For any $u\in\mathcal{S}(\mathbb{R}^d_{\theta}),$ we have where $G_t$ is the Gaussian function defined by def-Gaussian.

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 45 more