Global existence and blow-up of solutions for coupled bi-harmonic nonlinear wave equations

Abstract

Abstract This work studies the coupled nonlinear fourth-order wave system u ¨ i + Δ 2 ⁢ u i + u i = ± ( ∑ 1 ≤ j ≤ m a i ⁢ j ⁢ | u j | p ) ⁢ | u i | p - 2 ⁢ u i . \ddot{u}_{i}+\Delta^{2}u_{i}+u_{i}=\pm\bigg{(}\sum_{1\leq j\leq m}a_{ij}\lvert u% _{j}\rvert^{p}\biggr{)}\lvert u_{i}\rvert^{p-2}u_{i}. The main goal is to develop a local theory in the energy space and to investigate some issues of the global theory. Indeed, using a standard contraction argument coupled with Strichartz estimates, one obtains a local solution in the inhomogeneous Sobolev space ( H 1 ) m {(H^{1})^{m}} for the energy sub-critical regime. Then the local solution extends to a global one in the attractive regime; also in the energy critical case there is a global solution with small data. For a repulsive source term, by using the potential well theory with a concavity argument, the local solution may concentrate in finite time or extend to a global one. Finally, in the inter-critical regime, one proves the existence of infinitely many non-global solutions with data near to the stationary solution. Here, one needs to deal with the coupled source term which gives some technical restrictions such as p ≥ 2 {p\geq 2} in order to avoid a singularity. This assumption in the inter-critical regime gives a restriction on the space dimension.