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Journals / Contemporary Analysis and Applied Mathematics / 2013 / Cilt: 1 - Sayı: 2

Sönümleme ve kaynak terimi içeren lineer olmayan yüksek mertebeden Kirchhoff-tipi denklemlerin çözümlerinin üniform çöküşü ve patlaması

Uniform decay and blow up of solutions for a system of nonlinear higher-order Kirchhoff-type equations with damping and source terms

Erhan PİŞKİN
Necat POLAT
Pages
181–199
Publication Date
2022-10-11
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Also, we prove the blow up of the solution with negative initial energy by using the technique of [1] with a modification in the energy functional due to the different nature of the problems.

📌 3 Key Takeaways

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Objective & Hypothesis:This work studies the initial boundary value problem for the Kirchhoff- type equations    utt + M (∫ Ω (−△) m 2 u2 dx) (−△) m u + |ut| p−1 ut = f1 (u, v), vtt + M (∫ Ω (−△) m 2 v2 dx) (−△) m v + |vt| q−1 vt = f2 (u, v).
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Key Findings & Data:Under suitable conditions decay estimates of the solution are proved by using Nakao s inequality.
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Conclusion & Impact:Also, we prove the blow up of the solution with negative initial energy by using the technique of [1] with a modification in the energy functional due to the different nature of the problems.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

This work studies the initial boundary value problem for the Kirchhoff- type equations    utt + M (∫ Ω (−△) m 2 u2 dx) (−△) m u + |ut| p−1 ut = f1 (u, v), vtt + M (∫ Ω (−△) m 2 v2 dx) (−△) m v + |vt| q−1 vt = f2 (u, v). Under suitable conditions decay estimates of the solution are proved by using Nakao s inequality. Also, we prove the blow up of the solution with negative initial energy by using the technique of [1] with a modification in the energy functional due to the different nature of the problems.

Özet

Bu çalışmada Kirchhoff-tipi denklemler için başlangıç sınır değer problemi utt + M(∫Ω(−△)m2 u2 dx)(−△)m u + |ut|p−1ut = f1 (u, v),vtt + M(∫Ω(−△) m2 v2dx)(−△)m v + |vt|q−1vt = f2 (u, v) çalışılmıştır. Nakaonun eşitsizliği kullanılarak uygun koşullar altında çözümün çöküş yaklaşımları ispatlanmıştır. Ayrıca [1]da verilen teknikteki enerji fonksiyonelinin problemin farklı yapılarına göre modifikasyonunu kullanarak negatif başlangıç enerjisi ile çözümün patlaması ispatlanmıştır.

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Cite This Article

PİŞKİN, E., POLAT, N. (2022). Sönümleme ve kaynak terimi içeren lineer olmayan yüksek mertebeden Kirchhoff-tipi denklemlerin çözümlerinin üniform çöküşü ve patlaması. Contemporary Analysis and Applied Mathematics, Cilt: 1 - Sayı: 2, 181–199. https://ajindex.com/articles/476499

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