Satsanit, Wanchak; Kananthai, Amnuay Ultra-hyperbolic wave operator related to nonlinear wave equation. (English) Zbl 1242.35178 Int. J. Contemp. Math. Sci. 5, No. 1-4, 103-116 (2010). Authors’ abstract: “In this paper, we study the generalized wave equation of the form \[ \frac{\partial^2}{\partial t^2}u(x,t)+c^2 F u(x,t)=f(x,t,u(x,t)) \] with the initial conditions \[ u(x,0)=f(x),\quad\frac{\partial}{\partial t}u(x,0)=g(x), \] where \((x,t)\in \mathbb{R}^n\times [0,\infty)\), \(\mathbb{R}^n\) is the \(n\)-dimensional Euclidean space, \(F\) is the ultra-hyperbolic operator defined by \[ F=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\cdots+\frac{\partial^2}{\partial x_p^2}-\frac{\partial^2}{\partial x_{p+1}^2}-\frac{\partial^2}{\partial x_{p+2}^2}-\cdots-\frac{\partial^2}{\partial x_{p+q}^2}, \] \(p+q=n\), \(c\) is a positive constant, \(f\) and \(g\) are continuous and absolutely integrable functions. By \(\epsilon\)-approximation we also obtain the asymptotic solution \(u(x,t)=O(\epsilon^{-n})\ast f(x,t,u(x,t))\).” Reviewer: David Jornet (Valencia) MSC: 35L82 Pseudohyperbolic equations 47F05 General theory of partial differential operators 35L15 Initial value problems for second-order hyperbolic equations 35L71 Second-order semilinear hyperbolic equations Keywords:generalized wave equation; Fourier transform; tempered distribution PDFBibTeX XMLCite \textit{W. Satsanit} and \textit{A. Kananthai}, Int. J. Contemp. Math. Sci. 5, No. 1--4, 103--116 (2010; Zbl 1242.35178) Full Text: Link