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

DOI: 10.18173/2354-1059.2022-0018

Tóm tắt

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