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

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)