Global existence and blow-up of solutions to the double nonlinear porous medium equation

Bolys Sabitbek; Berikbol T. Torebek

doi:10.3934/dcds.2023124

Article Contents

2024, Volume 44, Issue 3: 743-767. Doi: 10.3934/dcds.2023124

This issue Previous Article Uniqueness and regularity of weak solutions of a fluid-rigid body interaction system under the Prodi-Serrin condition Next Article Continuum-wise hyperbolic homeomorphisms on surfaces

Global existence and blow-up of solutions to the double nonlinear porous medium equation

Bolys Sabitbek^{1, 2,} and
Berikbol T. Torebek^{2, 3, ,}

1.
School of Mathematical Sciences, Queen Mary University of London, United Kingdom
2.
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
3.
Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium

^*Corresponding author: Berikbol T. Torebek
^*Corresponding author: Berikbol T. Torebek

Received: June 2023

Revised: September 2023

Early access: October 2023

Published: March 2024

This research has been funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19676817). The second author was supported by FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations. No new data was collected or generated during the course of this research.

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Abstract

In this study, we examine a double nonlinear porous medium equation subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial energy. By introducing a set of potential wells, we construct invariant sets of solutions for the double nonlinear porous medium equation. For subcritical and critical initial energy scenarios, we derive the global existence and asymptotic behavior of weak solutions, as well as blow-up phenomena occurring within a finite time for the positive solution to the double nonlinear porous medium equation.

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Mathematics Subject Classification: Primary: 35K92, 35B44; Secondary: 35A01.

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