On The Regularizing Effect Of Nonlinear Damping In Hyperbolic Equations In Four-dimensional Space

Global well-posedness in H2(R4)× H1(R4)is shown for the initial value problem of nonlinear wave equations which as follows:? u + f(u)+ g(ut)= 0,t ? R+,x ? R4,u(0)= u0,ut(0)= u1,x ? R4,The conditions on nonlinear damping g(ut)are for all u:g(u)? C1(R),g(u)u?cm|u|m+1,|g(u)|? dm|u|m,g'(u)?0,and the defocusing nonlinear f(u)satisfies that for all u:f(u)? C2(R),f(0)=0,f'(u)? 0,f'(u)+ am? bmf"(u)sign(u)? 0.The result not only applies to polynomial functions,but also applies to certain exponential functi-ons,such as f(u)= sinh u.The main tool is Lions and Strauss' monotonicity method.It is observed that the nonlinear damping gives rise to a new monotone estimate which is play a key role to limit process.Moreover,global existence of solutions in H1(R4)× L2(R4)is shown for the same equation in the critical case f(u)= u3 and g(ut)= |ut|1/2ut.The main tool is a new estimate for the solution of the approximation problem,the existence of the solution of the original problem is proved by the limiting process.