The Cauchy Problem For A Kind Of 2m-Order Hyperbolic Equations

In this paper, we stady theCauchy problem of a kind of nonlinear 2m-order (m ≥2) hyperbolic equations: {u_n+(-1)^m â–³^m u+u=F(u) . we obtain decay estimates andspace-time interability estimates on solution to the homogeneous linear equation and smoothness and existence on the weak solutions to the non-homogeneous linear equations. We also use these estimates to study the global existence and the uniquness of the classical solution to the nonlinear equation.In section 2 we prove the L^p - L^q estimate of the homogeneous linear equation following the ideas used by Marshall and Strauss to obtain the estimates for the Klein-Gordon Equations. We find estimates on an analytic family ofoperators and use complex interpolation. The main L^p - L^q estimate (Theorem 2.3) states that solution of the linear equation with the initialdata (u₀,u₁)∈W^s+m,p'(Rⁿ)(?)W^s,p'(Rⁿ) are bound in W^s,p (Rⁿ) for all time and theirnorm in this space decays at the rate (1 + t)^{n/(mp)-n/(2m)} where p satisfy 2≤p≤2^*.In the sections 3.we prove the space-time integrablity estimate for the solution of linear equation.The result follows using techniques similar to those in section2,only with a careful analysis of the dependence of Bessel function of complex order.The space-time integrablity estimate(Theorem 3.3)states that solution of linear equation with initial data in the energy spaceZ≡H^(l+1)m(Rⁿ)(?)H^ml(Rⁿ) lie in the space W_p^l,ml(Rⁿ⁺¹) for all p satisfies...