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Ultra-hyperbolic wave operator related to nonlinear wave equation. (English) Zbl 1242.35178

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 \(u(x,t)=O(\epsilon^{-n})\ast f(x,t,u(x,t))\).”

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
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