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

Abstract

We investigate the asymptotic behavior of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff type $$\displaystyle u_{tt} -\phi (x){\| \nabla u(t)\|}^2 \varDelta u + \delta u_t = {|u|}^3 u, \hspace {2 mm} x\in R^N, \hspace {2 mm} t\geq 0, $$ with initial conditions u(x, 0) = u0(x) and ut(x, 0) = u1(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(u0, u1), which corresponds to the problem, is nonnegative and small, there exists a unique global solution in time in the space X0 =: D(A) × D1, 2(RN). When the initial energy E(u0, u1) 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 \(X_1 =: D^{1,2} (R^N) \times L^{2}_{g} (R^N)\) 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 $$\displaystyle u_{tt} = -{\| \mathrm {A}^{1/2} u \|}^{2}_{H} \mathrm {A} u + f(u), \hspace {2 mm} x\in R^N, \hspace {2 mm} t\geq 0, $$ 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) < 0, the solution is asymptotically stable.