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- W4300485788 abstract "In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$u_{tt}-left(a int_Omega |nabla u|^2 dif x +bright)Delta u = lambda u+ |u|^{p-1}u ,$$ where $a$, $b>0$, $p>1$, $lambda in mathbb{R}$ and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of $a$, $b$, $lambda$ and the initial data for the following cases: (i) $1<p<3$, (ii) $p=3$ and $a>1/Lambda$, (iii) $p=3$, $a leq 1/Lambda$ and $lam <blam_1$, (iv) $p=3$, $a < 1/Lambda$ and $lam >blam_1$, (v) $p>3$ and $lamleq blam_1$, (vi) $p>3$ and $lam> blam_1$, where $lam_1 = infleft{|nabla u|^2_2 :~ uin H^1_0(Omega) {rm and} |u|_2 =1right}$, and $Lambda = infleft{|nabla u|^4_2 :~ uin H^1_0(Omega) {rm and} |u|_4 =1right}$. Moreover, we prove the invariance of some stable and unstable sets of the solution for suitable $a$, $b$ and $lam$, and give the sufficient conditions of initial data to generate a vacuum region of the solution. Due to the nonlocal effect caused by the nonlocal integro-differential term, we show many interesting differences between the blow-up phenomenon of the problem for $a>0$ and $a=0$." @default.
- W4300485788 created "2022-10-03" @default.
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- W4300485788 date "2021-01-15" @default.
- W4300485788 modified "2023-10-16" @default.
- W4300485788 title "Dynamics of nonlinear hyperbolic equations of Kirchhoff type" @default.
- W4300485788 doi "https://doi.org/10.48550/arxiv.2101.06012" @default.
- W4300485788 hasPublicationYear "2021" @default.
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