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arxiv: 2605.03657 · v1 · submitted 2026-05-05 · 🧮 math.AP

Heat equations driven by mixed local-nonlocal operators with exponential nonlinearity

Dharmendra Kumar Chaurasia , Ahmad Z. Fino , Vishvesh Kumar This is my paper

Pith reviewed 2026-05-07 15:16 UTC · model grok-4.3

classification 🧮 math.AP
keywords heat equationmixed local-nonlocal operatorexponential nonlinearityOrlicz spacewell-posednessdecay estimatesfractional LaplacianCauchy problem
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The pith

Local solutions exist for mixed heat equation with exponential nonlinearity

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the Cauchy problem for a heat equation driven by the mixed operator combining the local Laplacian and a fractional Laplacian, with a nonlinearity that grows exponentially at large values and vanishes at zero. It establishes local well-posedness in a suitable Orlicz space when the growth is of the form e to the power of absolute value to p for p greater than 1. For small initial data, global solutions exist when the nonlinearity behaves like a power near the origin, and these solutions show large-time decay in Lebesgue spaces whose rate is fixed by that near-origin power. The results indicate that the mixed operator produces asymptotic behavior that combines aspects of purely local and purely nonlocal diffusion.

Core claim

For the Cauchy problem partial_t u + L u = f(u) with L equal to minus Delta plus the fractional Laplacian to the s, s in (0,1), and f exhibiting exponential growth at infinity with f(0)=0, local well-posedness holds in an appropriate Orlicz space. Global solutions exist for small initial data when f satisfies a power-type growth condition near zero. Large-time decay estimates in Lebesgue spaces are derived, with the decay rate determined by the behavior of f near the origin, thereby bridging the asymptotic theories of local and nonlocal diffusions.

What carries the argument

The mixed local-nonlocal operator L equals minus Delta plus (-Delta)^s for s in (0,1), acting together with fixed-point arguments in Orlicz spaces on the exponential nonlinearity f with f(0)=0.

If this is right

  • Local existence holds for nonlinearities with growth faster than any polynomial.
  • Global solutions for small data have decay rates controlled exclusively by the power m near zero.
  • The large-time decay in L^q spaces interpolates between the rates known for the pure heat equation and the fractional heat equation.
  • The mixed operator produces a unique asymptotic transition that depends on the origin behavior of f.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Orlicz-space approach may extend directly to other super-exponential nonlinearities or to mixed operators with variable fractional order.
  • The dominance of near-origin behavior in the decay suggests that numerical tests with different m could confirm the predicted transition between local and nonlocal regimes.
  • Similar techniques could yield blow-up criteria for large data by tracking the Orlicz norm growth.

Load-bearing premise

The nonlinearity f must satisfy exponential growth at infinity and power growth near zero while the linear mixed operator must be well-posed in the chosen Orlicz and Lebesgue spaces.

What would settle it

A concrete large initial datum in the Orlicz space for which no local solution exists, or numerical observation of decay rates that fail to depend on the near-zero power m.

read the original abstract

We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-\Delta+(-\Delta)^s$, $s\in(0,1)$, with exponential nonlinearity \[ \partial_tu(x,t)+\mathcal{L}u(x,t)=f(u(x,t)), \qquad (x,t)\in \mathbb{R}^{d}\times(0,\infty), \] where $f:\mathbb{R}\to\mathbb{R}$ exhibits exponential growth at infinity and satisfies $f(0)=0$. We establish local well-posedness in a suitable Orlicz space in the case where $f(u)\sim e^{|u|^p}$ as $|u|\to\infty$, with $p>1$. We further prove the existence of global solutions for small initial data under the assumption that $f$ satisfies the growth condition $|f(u)|\sim |u|^m$ near the origin. Moreover, we derive large-time decay estimates in Lebesgue spaces, showing that the behavior of the nonlinearity near the origin determines the decay rate of solutions and highlights a unique asymptotic transition that bridges local and non-local diffusion theories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the Cauchy problem for the heat equation ∂_t u + ℒu = f(u) on ℝ^d × (0,∞), where ℒ = -Δ + (-Δ)^s (s ∈ (0,1)) is a mixed local-nonlocal operator and f exhibits exponential growth at infinity with f(0)=0. It claims local well-posedness in a suitable Orlicz space when f(u) ∼ e^{|u|^p} as |u|→∞ (p>1), global existence for small initial data when |f(u)| ∼ |u|^m near the origin, and large-time decay estimates in Lebesgue spaces whose rate is determined by the near-origin behavior of f, thereby bridging local and nonlocal diffusion asymptotics.

Significance. If the central claims hold, the work would extend well-posedness and decay results for pure local or fractional heat equations to the mixed operator setting, with the Orlicz-space treatment of exponential nonlinearities and the identification of an asymptotic transition governed by local behavior near zero offering a concrete link between the two diffusion regimes. The approach via mild solutions and fixed-point arguments is standard for such problems and could serve as a template for related mixed-operator nonlinearities.

major comments (2)
  1. [Linear estimates / Section 2] The local well-posedness argument (abstract and the mild formulation u(t) = e^{-tℒ}u_0 + ∫_0^t e^{-(t-s)ℒ}f(u(s)) ds) requires that the semigroup generated by ℒ maps the chosen Orlicz space into itself with bounds compatible with the Nemytskii operator induced by the exponential nonlinearity. While the symbol |ξ|^2 + |ξ|^{2s} yields analytic semigroups on L^p (1<p<∞), the transfer of kernel estimates or extrapolation to the non-reflexive Orlicz space with exponential Young function is not automatic and receives no explicit justification or reference in the linear-theory section.
  2. [Decay estimates / Section 4] The large-time decay estimates in Lebesgue spaces are asserted to be controlled by the power-like behavior |f(u)| ∼ |u|^m near the origin and to exhibit a 'unique asymptotic transition' between local and nonlocal regimes. However, the proof sketch does not contain a direct comparison of the decay rates with the pure Laplacian (m-dependent) and pure fractional Laplacian cases, leaving the bridging claim without a quantitative anchor.
minor comments (2)
  1. [Abstract] The precise Orlicz space (Young function and associated norm) is referred to only as 'suitable' in the abstract; an explicit definition should appear at the first use in the introduction or linear-theory section.
  2. [Global existence / Section 3] The growth condition |f(u)| ∼ |u|^m near the origin is used for global existence but the admissible range of m relative to the dimension d and the fractional order s is not stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate revisions to strengthen the linear theory and decay analysis.

read point-by-point responses

  1. Referee: [Linear estimates / Section 2] The local well-posedness argument (abstract and the mild formulation u(t) = e^{-tℒ}u_0 + ∫_0^t e^{-(t-s)ℒ}f(u(s)) ds) requires that the semigroup generated by ℒ maps the chosen Orlicz space into itself with bounds compatible with the Nemytskii operator induced by the exponential nonlinearity. While the symbol |ξ|^2 + |ξ|^{2s} yields analytic semigroups on L^p (1<p<∞), the transfer of kernel estimates or extrapolation to the non-reflexive Orlicz space with exponential Young function is not automatic and receives no explicit justification or reference in the linear-theory section.

    Authors: We agree that the manuscript would benefit from explicit justification of the semigroup action on the Orlicz space. In the revised version we will add a subsection to Section 2 deriving the necessary kernel bounds for the mixed operator from its symbol and establishing the required extrapolation from L^p to the exponential Orlicz space, with references to the relevant literature on analytic semigroups in non-reflexive Orlicz spaces. This will make the fixed-point argument fully self-contained. revision: yes

  2. Referee: [Decay estimates / Section 4] The large-time decay estimates in Lebesgue spaces are asserted to be controlled by the power-like behavior |f(u)| ∼ |u|^m near the origin and to exhibit a 'unique asymptotic transition' between local and nonlocal regimes. However, the proof sketch does not contain a direct comparison of the decay rates with the pure Laplacian (m-dependent) and pure fractional Laplacian cases, leaving the bridging claim without a quantitative anchor.

    Authors: The referee correctly identifies that an explicit side-by-side comparison would strengthen the bridging claim. We will revise Section 4 to include a short comparative subsection that recalls the known m-dependent decay rates for the pure Laplacian and pure fractional Laplacian, then shows quantitatively that the mixed-operator decay coincides with the local rate (governed by the near-origin power m) while differing from the fractional rate, thereby anchoring the asserted asymptotic transition. revision: yes

Circularity Check

0 steps flagged

No circularity in the derivation chain

full rationale

The paper establishes local well-posedness for the nonlinear problem via the mild formulation and fixed-point arguments in Orlicz spaces, relying on the linear mixed operator generating an analytic semigroup (from its Fourier symbol) and standard embeddings/Nemytskii operator properties for the exponential nonlinearity. Global existence for small data and decay estimates follow directly from the assumed growth conditions on f near the origin and at infinity, without any reduction of outputs to fitted parameters, self-definitions, or load-bearing self-citations. The claims are self-contained against external semigroup theory benchmarks and do not rename known results or smuggle ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from functional analysis and PDE theory for the linear operator and function spaces; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math The mixed operator L generates a suitable semigroup on the chosen Orlicz and Lebesgue spaces.
    Invoked implicitly for well-posedness of the linear problem underlying the nonlinear analysis.
  • standard math Standard embeddings and inequalities hold for Orlicz spaces with exponential growth.
    Required for handling the exponential nonlinearity in the fixed-point argument.

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