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Heat equations associated to harmonic oscillator with exponential nonlinearity

Divyang G. Bhimani, Mohamed Majdoub, Ramesh Manna

TL;DR

This work analyzes the Cauchy problem for the heat equation with a fractional harmonic oscillator and an exponential nonlinearity, namely $\partial_t u + H^\beta u = f(u)$ with $H=-\Delta+|x|^2$, exploring solutions in Orlicz spaces $\exp L^p$ and $\exp L^p_0$. The authors establish local well-posedness in $\exp L^p_0$ for $0<\beta\le 1$, and show global existence of small-data solutions in $\exp L^p$ with precise decay rates $\|u(t)\|_{L^a} \lesssim t^{-\left(\frac{1}{m-1}-\frac{d}{2\beta a}\right)}$, reflecting the nonlinearity’s near-origin behavior. They also demonstrate nonexistence of local nonnegative classical solutions for certain initial data in $\exp L^p$, highlighting sharp limitations of positivity results in this setting. Overall, the paper extends NLH-type results to exponential nonlinearities under a harmonic-oscillator potential, using a careful Orlicz-space fixed-point framework and detailed semigroup estimates, with implications for self-trapped beam models and related PDEs.

Abstract

We investigate the Cauchy problem for a heat equation involving a fractional harmonic oscillator and an exponential nonlinearity. We establish local well-posedness within the appropriate Orlicz spaces. Through the examination of small initial data in suitable Orlicz spaces, we obtain the existence of global weak-mild solutions. Additionally, precise decay estimates are presented for large time, indicating that the decay rate is influenced by the nonlinearity's behavior near the origin. Moreover, we highlight that the existence of local nonnegative classical solutions is no longer guaranteed when certain nonnegative initial data is considered within the appropriate Orlicz space.

Heat equations associated to harmonic oscillator with exponential nonlinearity

TL;DR

This work analyzes the Cauchy problem for the heat equation with a fractional harmonic oscillator and an exponential nonlinearity, namely with , exploring solutions in Orlicz spaces and . The authors establish local well-posedness in for , and show global existence of small-data solutions in with precise decay rates , reflecting the nonlinearity’s near-origin behavior. They also demonstrate nonexistence of local nonnegative classical solutions for certain initial data in , highlighting sharp limitations of positivity results in this setting. Overall, the paper extends NLH-type results to exponential nonlinearities under a harmonic-oscillator potential, using a careful Orlicz-space fixed-point framework and detailed semigroup estimates, with implications for self-trapped beam models and related PDEs.

Abstract

We investigate the Cauchy problem for a heat equation involving a fractional harmonic oscillator and an exponential nonlinearity. We establish local well-posedness within the appropriate Orlicz spaces. Through the examination of small initial data in suitable Orlicz spaces, we obtain the existence of global weak-mild solutions. Additionally, precise decay estimates are presented for large time, indicating that the decay rate is influenced by the nonlinearity's behavior near the origin. Moreover, we highlight that the existence of local nonnegative classical solutions is no longer guaranteed when certain nonnegative initial data is considered within the appropriate Orlicz space.
Paper Structure (10 sections, 20 theorems, 135 equations)

This paper contains 10 sections, 20 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and key estimates
  3. Proof of Theorem \ref{['T1.1']}
  4. Proof of Theorem \ref{['Th1']}
  5. The case $d>\frac{2\beta p}{p-1}$
  6. The case $d<\frac{2\beta p}{p-1}$
  7. The case $d=\frac{2\beta p}{p-1}$
  8. Proof of Theorem \ref{['Th1']}
  9. Proof of Theorem \ref{['Thne']}
  10. Conclusion and Open Problems

Key Result

Theorem 1.1

Let $u_0 \in \exp L_0^p(\mathbb R^d)$ and $0< \beta \leq 1.$ Assume that $f$ satisfies nflw. Then there exists $T= T(u_0)>0$ and a unique mild solution $u \in C([0, T], \exp L^p_0(\mathbb R^d))$ to hexp.

Theorems & Definitions (38)

  • Theorem 1.1: Local well-posedness
  • Remark 1.1
  • Theorem 1.2: Global existence
  • Remark 1.2
  • Definition 1.1: $\exp L^p$ -classical solution
  • Theorem 1.3: Nonexistence
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1: Orlicz space
  • Lemma 2.1: Inclusion properties
  • ...and 28 more