Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain
In this paper, we consider a time fractional diffusion system with a nonlinear memory term in a bounded domain. We mainly prove some blow-up and global existence results for this problem. Moreover, we also give the decay estimates of the global solutions. Our proof relies on the eigenfunction method...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/1/56 |
| _version_ | 1797442308991352832 |
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| author | Quanguo Zhang Yaning Li |
| author_facet | Quanguo Zhang Yaning Li |
| author_sort | Quanguo Zhang |
| collection | DOAJ |
| description | In this paper, we consider a time fractional diffusion system with a nonlinear memory term in a bounded domain. We mainly prove some blow-up and global existence results for this problem. Moreover, we also give the decay estimates of the global solutions. Our proof relies on the eigenfunction method combined with the asymptotic behavior of the solution of a fractional differential inequality system, the estimates of the solution operators and the asymptotic behavior of the Mittag–Leffler function. In particular, we give the critical exponents of this problem in different cases. Our results show that, in some cases, whether one of the initial values is identically equal to zero has a great influence on blow-up and global existence of the solutions for this problem, which is a remarkable property of time fractional diffusion systems because the classical diffusion systems can not admit this property. |
| first_indexed | 2024-03-09T12:39:58Z |
| format | Article |
| id | doaj.art-86f0e5460efd4aa5a6a154899da4eed8 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T12:39:58Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-86f0e5460efd4aa5a6a154899da4eed82023-11-30T22:19:38ZengMDPI AGFractal and Fractional2504-31102023-01-01715610.3390/fractalfract7010056Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded DomainQuanguo Zhang0Yaning Li1Department of Mathematics, Luoyang Normal University, Luoyang 471022, ChinaCollege of Mathematics & Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, ChinaIn this paper, we consider a time fractional diffusion system with a nonlinear memory term in a bounded domain. We mainly prove some blow-up and global existence results for this problem. Moreover, we also give the decay estimates of the global solutions. Our proof relies on the eigenfunction method combined with the asymptotic behavior of the solution of a fractional differential inequality system, the estimates of the solution operators and the asymptotic behavior of the Mittag–Leffler function. In particular, we give the critical exponents of this problem in different cases. Our results show that, in some cases, whether one of the initial values is identically equal to zero has a great influence on blow-up and global existence of the solutions for this problem, which is a remarkable property of time fractional diffusion systems because the classical diffusion systems can not admit this property.https://www.mdpi.com/2504-3110/7/1/56time fractional diffusion systemblow-upglobal existencecritical exponentnonlinear memory |
| spellingShingle | Quanguo Zhang Yaning Li Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain Fractal and Fractional time fractional diffusion system blow-up global existence critical exponent nonlinear memory |
| title | Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain |
| title_full | Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain |
| title_fullStr | Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain |
| title_full_unstemmed | Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain |
| title_short | Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain |
| title_sort | threshold results for the existence of global and blow up solutions to a time fractional diffusion system with a nonlinear memory term in a bounded domain |
| topic | time fractional diffusion system blow-up global existence critical exponent nonlinear memory |
| url | https://www.mdpi.com/2504-3110/7/1/56 |
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