Global existence for certain fourth order evolution equations

Rafael Granero-Belinchón; Martina Magliocca

doi:10.3934/cpaa.2024060

Article Contents

2024, Volume 23, Issue 11: 1646-1660. Doi: 10.3934/cpaa.2024060

This issue Previous Article Next Article Existence of least energy solutions for a quasilinear Choquard equation

Global existence for certain fourth order evolution equations

Rafael Granero-Belinchón^1, , and
Martina Magliocca^2,

1.
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Spain
2.
Departamento de Análisis Matemático, Universidad de Sevilla, Spain

^*Corresponding author: Rafael Granero-Belinchón
^*Corresponding author: Rafael Granero-Belinchón

Received: December 2023

Revised: July 2024

Early access: August 2024

Published online: August 2024

R.G-B was supported by the project "Mathematical Analysis of Fluids and Applications" Grant PID2019-109348GA-I00 funded by MCIN/AEI/ 10.13039/501100011033 and acronym "MAFyA". This publication is part of the project PID2019-109348GA-I00 funded by MCIN/ AEI /10.13039/501100011033. R.G-B is also supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The BBVA Foundation accepts no responsibility for the opinions, statements, and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors.
The work of M.M. was partially supported by Grant RYC2021-033698-I, funded by the Ministry of Science and Innovation/State Research Agency/10.13039/501100011033 and by the European Union "NextGenerationEU/Recovery, Transformation and Resilience Plan".
Both authors are funded by the project "Análisis Matemático Aplicado y Ecuaciones Diferenciales" Grant PID2022-141187NB-I00 funded by MCIN /AEI /10.13039/501100011033 / FEDER, UE and acronym "AMAED". This publication is part of the project PID2022-141187NB-I00 funded by MCIN/ AEI /10.13039/501100011033.

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Abstract

In this paper we established three global in time results for two fourth order nonlinear parabolic equations. The first of such equations involved the Hessian and appeared in epitaxial growth. For such an equation, we gave conditions ensuring the global existence of the solution. For certain regime of the parameters, our size condition involved the norm in a critical space with respect to the scaling of the equation and improved previous existing results in the literature for this equation. The second of the equations under study was a thin film equation with a porous medium nonlinearity. For this equation, we established conditions leading to the global existence of the solution.

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Mathematics Subject Classification: 35K25, 35K55, 35K30, 35Q35, 35Q92.

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References

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