THE SEMILINEAR COUPLED SYSTEMS FOR THE EXTERNAL DAMPING MODELS WITH VARIABLE COEFFICIENTS

We present in this article some results on the global solvability with arbitrarily small data of the Cauchy problem for the following semilinear coupled system with a variable coefficient utt + a(x)(−∆)σu + ut = F(|D|αv, vt), vtt + a(x)(−∆)σv + vt = G(|D|αu, ut), u(0, x) = u0(x), ut(0, x) = u1(x), v(0, x) = v0(x), vt(0, x) = v1(x). The nonlinearities are of the form (F, G) = (||D|αv|p, ||D|αu|q), or (F, G) = (|vt|p, |ut|q), and the parameter σ satisfies σ ∈ (0, 1). We will show that the ”critical exponents” p, q for the small data global solvability have a close relation to the established exponents of the corresponding semilinear problems for the external damping equations.