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On certain integral equations related to nonlinear wave equations. (English) Zbl 0804.45001
This paper is concerned with the global in time existence for integral equations such as the following (1) \(u(x,t) = v(x,t) + L(F(u)) (x,t)\), where \(L(\varphi) (x,t) = A_ n \int^ t_ 0 (t - \tau) M(\varphi/x,t - \tau; \tau) d \tau\). Moreover, \(v'\) and \(F\) are given functions and \(A_ n\) is a given positive constant. \(L\) is a positive linear operator and \(M\) is defined by integrals: \(M(\varphi/x,r;t) = \int_{| w | = 1} \varphi (x + rw,t) dS_ w\), \(n = 2m + 1\); \(M(\varphi/x,r;t) = \int_{| \varepsilon | \leq 1} (\varphi (x + r \varepsilon,t)/(1- | \varepsilon |^ 2)^{1/2}) d \varepsilon\), \(n = 2m\).
The authors find that a solution \(u(x,t)\) to the integral equation (1) is a solution to the Cauchy problem for a nonlinear wave equation of the form \(\partial^ 2_ t u(x,t) - \Delta u(x,t) = F(u) (x,t) - H(x,t)\), \(u(x,0) = f(x)\), \(\partial_ t u(x,0) = g(x)\), where \(H(x,t) = 2(m - 1) A_ n \int^ t_ 0 M(\partial_ t (F(u))/x,t - \tau; \tau) d \tau\). The functions \(f\) and \(g\) occur in a Cauchy problem for a linear wave equation.
The global existence of \(C^ 1\) solutions to the integral equation (1) is proved provided a suitable norm of \(v\) is small and with some hypothesis on \(F\).

45G10 Other nonlinear integral equations
35L70 Second-order nonlinear hyperbolic equations
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