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Global existence and some qualitative properties of weak solutions for a class of pseudo-parabolic equations with a logarithmic nonlinearity in whole $\mathbb{R}^{N}$

Part of: Parabolic equations and systems Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  01 August 2025

Claudianor Alves  and
Tahir Boudjeriou
Claudianor Alves
Affiliation:
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, Paraíba 58429-970, Brazil (coalves@mat.ufcg.edu.br)
Tahir Boudjeriou*
Affiliation:
Department of Basic Teaching, Institute of Electrical & Electronic Engineering, University of Boumerdes, Boumerdes 35000, Algeria (t.boudjeriou@univ-boumerdes.dz) (corresponding author)
*
*Corresponding author.
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Abstract

In this paper, we study the Cauchy problem for pseudo-parabolic equations with a logarithmic nonlinearity. After establishing the existence and uniqueness of weak solutions within a suitable functional framework, we investigate several qualitative properties, including the asymptotic behaviour and blow-up of solutions as $t\to +\infty$. Moreover, when the initial data are close to a Gaussian function, we prove that these weak solutions exhibit either super-exponential growth or super-exponential decay.

Keywords

blow-up solutionglobal solutionpseudo-parabolic equationsglobal existenceblow-upasymptotic behaviorlogarithmic nonlinearity

MSC classification

Primary: 35K05: Heat equation
Secondary: 35J10: Schrödinger operator 35B44: Blow-up

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 44
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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