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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 $$F$$.

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