Solutions of Cauchy problem for multiple inhomogeneous wave equation

Abstract

We define a class of pseudo-differential operators in a completely new way, which is called the abstract operators and expounded systematically the theory of abstract operators. By combining abstract operators with the Laplace transform, we can apply the Laplace transform to any $n+1$ dimensional linear higher-order partial differential equations $P(\partial_x,\partial_t)u=f(x,t)$ directly, without using the Fourier transform. By making introduction of abstract operators $G(\partial_x,t):=\mathcal{L}^{-1}[1/P(\partial_x,s)]$, the analytic solutions of initial value problems are expressed in these abstract operators, including the multiple inhomogeneous wave equation associated with the shifted Laplace-Beltrami operator on real hyperbolic spaces. By writing abstract operators in this class into integral forms, the solutions in operator form are represented into integral forms. Thus the analytic solutions of Cauchy problem for the multiple wave equation on $\mathbb{R}^n$ can be represented in the integrations of some given functions, without using the traditional Fourier transform technique. As a further application, we study the solvability of initial-boundary value problem for the linear higher-order partial differential equations and deduce new distinguishable method associated with the second-order linear self-adjoint elliptic operators.