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arxiv: 2106.11620 · v3 · submitted 2021-06-22 · 🧮 math.AP

Blow-up results for a logarithmic pseudo-parabolic p(.)-Laplacian type equation

Belhaoues Razik , Umberto Biccari , Abita Rahmoune This is my paper

Pith reviewed 2026-05-24 13:36 UTC · model grok-4.3

classification 🧮 math.AP
keywords blow-uppseudo-parabolic equationvariable exponentslogarithmic nonlinearitydecay estimatesenergy methodfinite time blow-up
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The pith

Solutions to a logarithmic pseudo-parabolic p(·)-Laplacian equation blow up in finite time under suitable initial conditions, with an explicit upper bound on the blow-up time.

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

This paper studies an initial-boundary value problem for a mixed pseudo-parabolic equation with variable exponents and a logarithmic nonlinearity. Adapting the first-order differential inequality method to the energy functional produces a blow-up criterion together with an upper bound on the blow-up time. The authors further show that sufficiently small initial data prevent blow-up entirely and yield decay of the solution in the H01 norm. These results clarify the competition between the diffusion and the nonlinear source terms in equations whose exponents vary in space.

Core claim

By applying the first-order differential inequality method directly to the energy functional, the paper establishes a blow-up criterion for solutions of the equation u_t - Δu_t - div(|∇u|^{p(·)-2} ∇u) = |u|^{q(·)-2} u ln(|u|). An upper bound on the blow-up time follows from the same inequality. When the initial datum is small enough in the appropriate norm, blow-up is avoided and the H01(Ω) norm of the solution decays to zero as t → ∞.

What carries the argument

The first-order differential inequality method applied to the energy functional, which converts the PDE into a scalar inequality that controls the sign of the time derivative and produces the blow-up time bound.

If this is right

  • An explicit upper bound for the blow-up time is obtained whenever the blow-up criterion holds.
  • Blow-up is prevented when the initial datum is sufficiently small.
  • Decay estimates in the H01(Ω) norm hold as t → +∞ under the smallness condition.
  • The decay result is illustrated by a two-dimensional numerical simulation.

Where Pith is reading between the lines

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

  • The same inequality technique might be tested on related equations whose nonlinearity is not logarithmic.
  • The derived time bound could be compared against computed blow-up times to check sharpness.
  • The decay result implies that the zero solution is asymptotically stable for small perturbations in this variable-exponent setting.

Load-bearing premise

The variable exponents p(·) and q(·) must obey regularity conditions that let the first-order differential inequality be applied directly to the energy without extra obstructions.

What would settle it

A concrete numerical solution or explicit example whose blow-up time exceeds the derived upper bound, or whose H01 norm fails to decay when the initial datum satisfies the smallness condition.

Figures

Figures reproduced from arXiv: 2106.11620 by Abita Rahmoune, Belhaoues Razik, Umberto Biccari.

Figure 4
Figure 4. Figure 4: Time evolution of the solution of (5.1) for t ∈ [0, 1]. Acknowledgments This work is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon), by the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-18-1-0242, by Grants MTM2017-92996-C2- 1-R COSNET and PID2020-112617GB-C22 KILEARN of M… view at source ↗
read the original abstract

In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-\Delta u_t-\mbox{div}\left(\left\vert \nabla u\right\vert^{p(.)-2}\nabla u\right) =|u|^{q(.)-2}u\ln(|u|), \quad (x,t)\in\Omega\times(0,+\infty),$$ where $\Omega\subset\mathbb{R}^n$ is a bounded and regular domain, and the variable exponents $p(.)$ and $q(.)$ satisfy suitable regularity assumptions. By adapting the first-order differential inequality method, we establish a blow-up criterion for the solutions and obtain an upper bound for the blow-up time. In a second moment, we show that blow-up may be prevented under appropriate smallness conditions on the initial datum, in which case we also establish decay estimates in the $H_0^1(\Omega)$-norm as $t\to+\infty$. This decay result is illustrated by a two-dimensional numerical example.

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

0 major / 2 minor

Summary. The manuscript considers an initial-boundary value problem for the mixed pseudo-parabolic p(·)-Laplacian equation with logarithmic nonlinearity u_t - Δu_t - div(|∇u|^{p(·)-2} ∇u) = |u|^{q(·)-2} u ln(|u|). By adapting the first-order differential inequality method to an energy functional that incorporates the variable exponents, the authors establish a blow-up criterion together with an explicit upper bound on the blow-up time. Under smallness assumptions on the initial datum they prove global existence and obtain decay estimates for the H_0^1(Ω)-norm as t → ∞; the decay result is illustrated by a two-dimensional numerical example.

Significance. The work extends the first-order differential inequality technique, previously applied to constant-exponent pseudo-parabolic problems, to the variable-exponent setting with a logarithmic source. If the technical steps are justified under the stated regularity hypotheses on p(·) and q(·), the results supply new blow-up and decay criteria for a class of equations arising in non-standard growth models. The combination of an explicit blow-up time bound, a small-data global-existence theorem, and a numerical illustration is a positive feature.

minor comments (2)
  1. [Abstract] Abstract, paragraph 2: the phrase “suitable regularity assumptions” on p(·) and q(·) is too vague for a reader to verify applicability of the differential inequality method; the precise hypotheses (log-Hölder continuity, bounds away from 1 and ∞, etc.) should be stated explicitly already in the introduction or in a dedicated preliminary section.
  2. [Numerical example] The numerical example is mentioned only in the abstract; the main text should include at least a brief description of the discretization scheme, mesh size, and the specific variable exponents used, so that the illustration can be assessed for consistency with the analytic decay rate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript, the positive evaluation of its significance, and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard inequality method to energy functional

full rationale

The paper adapts the first-order differential inequality method to the energy functional of the given PDE with variable exponents p(·), q(·) and logarithmic source. The blow-up criterion, upper bound on blow-up time, and decay estimates under small initial data are derived as consequences of this inequality under stated regularity assumptions on the exponents. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the numerical example is presented only as illustration. The derivation remains self-contained against the PDE and the classical method.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; the ledger is populated from the method name and the stated regularity assumptions on the exponents. No free parameters or invented entities are visible.

axioms (1)
  • domain assumption Variable exponents p(·), q(·) satisfy suitable regularity assumptions allowing the differential inequality method to close.
    Invoked in abstract paragraph 2 to justify application of the method.

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