Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term
In this paper, we address the following initial value problem\[\begin{array}{ll}\hbox{\(u_t=\int_{\Omega}J(x-y)(u(y, t)-u(x, t)){\rm d}y+f(u(x, t))\quad \mbox{in}\quad \overline{\Omega}\times(0,T)\),} \\\hbox{\(u(x,0)=u_{0}(x)\geq 0\quad \mbox{in}\quad \overline{\Omega}\),} \\\end{array}\]where \(\O...
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| Format: | Article |
| Language: | English |
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Publishing House of the Romanian Academy
2012-08-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
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| Online Access: | https://www.ictp.acad.ro/jnaat/journal/article/view/975 |
| _version_ | 1811294069657174016 |
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| author | Halima Nachid |
| author_facet | Halima Nachid |
| author_sort | Halima Nachid |
| collection | DOAJ |
| description | In this paper, we address the following initial value problem\[\begin{array}{ll}\hbox{\(u_t=\int_{\Omega}J(x-y)(u(y, t)-u(x, t)){\rm d}y+f(u(x, t))\quad \mbox{in}\quad \overline{\Omega}\times(0,T)\),} \\\hbox{\(u(x,0)=u_{0}(x)\geq 0\quad \mbox{in}\quad \overline{\Omega}\),} \\\end{array}\]where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(f: (-\infty, b)\rightarrow (0, \infty)\) is a \(C^1\) convex nondecreasing function, \(\lim_{s\rightarrow b^{-}}f(s)=\infty\), \(\int^{\infty}\tfrac{{\rm d}\sigma}{f(\sigma)}<\infty\), with \(b\) a positive constant, \(J:\mathbb{R}^N\rightarrow \mathbb{R}\) is a kernel which is measurable, nonnegative and bounded in \(\mathbb{R}^N\). Under some conditions, we show that the solution of a perturbed form of the above problem quenches in a finite time and estimate its quenching time. We also prove the continuity of the quenching time as a function of the initial datum. Finally, we give some numerical results to illustrate our analysis. |
| first_indexed | 2024-04-13T05:10:55Z |
| format | Article |
| id | doaj.art-77f053ac4a25452d9ed2bfd5983adfdf |
| institution | Directory Open Access Journal |
| issn | 2457-6794 2501-059X |
| language | English |
| last_indexed | 2024-04-13T05:10:55Z |
| publishDate | 2012-08-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj.art-77f053ac4a25452d9ed2bfd5983adfdf2022-12-22T03:01:01ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2012-08-01412Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction termHalima Nachid0Université d'Abobo-AdjaméIn this paper, we address the following initial value problem\[\begin{array}{ll}\hbox{\(u_t=\int_{\Omega}J(x-y)(u(y, t)-u(x, t)){\rm d}y+f(u(x, t))\quad \mbox{in}\quad \overline{\Omega}\times(0,T)\),} \\\hbox{\(u(x,0)=u_{0}(x)\geq 0\quad \mbox{in}\quad \overline{\Omega}\),} \\\end{array}\]where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(f: (-\infty, b)\rightarrow (0, \infty)\) is a \(C^1\) convex nondecreasing function, \(\lim_{s\rightarrow b^{-}}f(s)=\infty\), \(\int^{\infty}\tfrac{{\rm d}\sigma}{f(\sigma)}<\infty\), with \(b\) a positive constant, \(J:\mathbb{R}^N\rightarrow \mathbb{R}\) is a kernel which is measurable, nonnegative and bounded in \(\mathbb{R}^N\). Under some conditions, we show that the solution of a perturbed form of the above problem quenches in a finite time and estimate its quenching time. We also prove the continuity of the quenching time as a function of the initial datum. Finally, we give some numerical results to illustrate our analysis.https://www.ictp.acad.ro/jnaat/journal/article/view/975nonlocal diffusionquenchingcontinuitynumerical quenching timereaction-diffusion equation |
| spellingShingle | Halima Nachid Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term Journal of Numerical Analysis and Approximation Theory nonlocal diffusion quenching continuity numerical quenching time reaction-diffusion equation |
| title | Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term |
| title_full | Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term |
| title_fullStr | Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term |
| title_full_unstemmed | Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term |
| title_short | Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term |
| title_sort | remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term |
| topic | nonlocal diffusion quenching continuity numerical quenching time reaction-diffusion equation |
| url | https://www.ictp.acad.ro/jnaat/journal/article/view/975 |
| work_keys_str_mv | AT halimanachid remarksonthequenchingestimateforanonlocaldiffusionproblemwithareactionterm |