Consider a system of nonlinear wave equations (e)2t-c2i△xui=Fi(u,(e)u,(e)x(e)u) in (0,∞)×(R)3 for i=1,┈,m,where Fi(i=1,┈,m) are smooth functions of degree 2 near the origin of their arguments, and u=(u1,┈,um...Consider a system of nonlinear wave equations (e)2t-c2i△xui=Fi(u,(e)u,(e)x(e)u) in (0,∞)×(R)3 for i=1,┈,m,where Fi(i=1,┈,m) are smooth functions of degree 2 near the origin of their arguments, and u=(u1,┈,um),while (e)u and (e)x(e)u represent the first and (second derivatives of u, respectively. In this paper, the author presents a new class of nonlinearity for which the global existence of small solutions is ensured. For example, global existence of small solutions for((e)2t- c21Δx)u1 = u2((e)tu2) + arbitrary cubic terms,((e)2t - c22Δx)u2=u1((e)tu2) + ((e)tu1)u2 + arbitrary cubic termswill be established, provided that c21 ≠ c22.展开更多
文摘Consider a system of nonlinear wave equations (e)2t-c2i△xui=Fi(u,(e)u,(e)x(e)u) in (0,∞)×(R)3 for i=1,┈,m,where Fi(i=1,┈,m) are smooth functions of degree 2 near the origin of their arguments, and u=(u1,┈,um),while (e)u and (e)x(e)u represent the first and (second derivatives of u, respectively. In this paper, the author presents a new class of nonlinearity for which the global existence of small solutions is ensured. For example, global existence of small solutions for((e)2t- c21Δx)u1 = u2((e)tu2) + arbitrary cubic terms,((e)2t - c22Δx)u2=u1((e)tu2) + ((e)tu1)u2 + arbitrary cubic termswill be established, provided that c21 ≠ c22.
