Existence of Global Solutions and Stability Results for a Nonlinear Wave Problem in Unbounded Domains

P. Papadopoulos (), N. L. Matiadou (), S. Fatouros () and G. Xerogiannakis ()
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P. Papadopoulos: University of West Attica
N. L. Matiadou: University of West Attica
S. Fatouros: University of West Attica
G. Xerogiannakis: University of West Attica

A chapter in Mathematical Analysis in Interdisciplinary Research, 2021, pp 589-598 from Springer

Abstract: Abstract We investigate the asymptotic behavior of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff type u t t − ϕ ( x ) ∥ ∇ u ( t ) ∥ 2 Δ u + δ u t = | u | 3 u , x ∈ R N , t ≥ 0 , $$\displaystyle u_{tt} -\phi (x){\| \nabla u(t)\|}^2 \varDelta u + \delta u_t = {|u|}^3 u, \hspace {2 mm} x\in R^N, \hspace {2 mm} t\geq 0, $$ with initial conditions u(x, 0) = u 0(x) and u t(x, 0) = u 1(x), in the case where N ≥ 3, δ > 0, and (ϕ(x))−1 = g(x) is a positive function lying in LN∕2(RN) ∩ L∞(RN). It is proved that when the initial energy E(u 0, u 1), which corresponds to the problem, is nonnegative and small, there exists a unique global solution in time in the space X 0 =: D(A) × D1, 2(RN). When the initial energy E(u 0, u 1) is negative, the solution blows up in finite time. For the proofs, a combination of the modified potential well method and the concavity method is used. Also, the existence of an absorbing set in the space X 1 = : D 1 , 2 ( R N ) × L g 2 ( R N ) $$X_1 =: D^{1,2} (R^N) \times L^{2}_{g} (R^N)$$ is proved and that the dynamical system generated by the problem possess an invariant compact set A in the same space. Finally, for the generalized Kirchhoff’s string problem with no dissipation u t t = − ∥ A 1 ∕ 2 u ∥ H 2 A u + f ( u ) , x ∈ R N , t ≥ 0 , $$\displaystyle u_{tt} = -{\| \mathrm {A}^{1/2} u \|}^{2}_{H} \mathrm {A} u + f(u), \hspace {2 mm} x\in R^N, \hspace {2 mm} t\geq 0, $$ with the same hypotheses as above, we study the stability of the trivial solution u ≡ 0. It is proved that if f′(0) > 0, then the solution is unstable for the initial Kirchhoff’s system, while if f′(0)

Keywords: Quasilinear hyperbolic equations; Global solution; Blow up; Dissipation; Potential well; Concavity method; Unbounded domains; Kirchhoff strings; Generalized Sobolev spaces; Weighted Lp spaces (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-84721-0_26

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DOI: 10.1007/978-3-030-84721-0_26

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