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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
It is also shown that regarding arbitrary positive initial energy and suitable range of variable exponents, solutions blow-up in a finite time.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper we consider $r(x)-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\big(a+b\int_{\Omega}\frac{1}{r(x)}|\nabla u|^{r(x)}dx\big)\Delta_{r(x)}u+\beta u_{t}=|u|^{p(x)-2}u, $$ associated with initial and Dirichlet boundary conditions.
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Key Findings & Data:Under appropriate conditions on $r(.)$ and $p(.)$, stability result along the solution energy is proved.
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Conclusion & Impact:It is also shown that regarding arbitrary positive initial energy and suitable range of variable exponents, solutions blow-up in a finite time.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Kirchhoff equation · stability result · variable exponents
Abstract
In this paper we consider $r(x)-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\big(a+b\int_{\Omega}\frac{1}{r(x)}|\nabla u|^{r(x)}dx\big)\Delta_{r(x)}u+\beta u_{t}=|u|^{p(x)-2}u, $$ associated with initial and Dirichlet boundary conditions. Under appropriate conditions on $r(.)$ and $p(.)$, stability result along the solution energy is proved. It is also shown that regarding arbitrary positive initial energy and suitable range of variable exponents, solutions blow-up in a finite time.
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Cite This Article
SHAHROUZİ, M., FERREIRA, J. (2021). A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy. Communications in Advanced Mathematical Sciences, Cilt: 4 - Sayı: 4, 208–216. https://ajindex.com/articles/748231