Some existence results for a nonlinear hyperbolic integrodifferential equation with singular kernel
The author studies the nonlinear Volterra integro-differential equation \[ u_ t(t,x)=\int^ t_ 0a(t-s)b(u_ x(s,x))_ xds+f(t,x),\quad t\geq 0,\;x\in\mathbb{R} \] \(u(0,x)=u_ 0(x)\) where \(a\) is of strong positive type and \(b\in C^ 3(\mathbb{R})\), \(b(0)=0\), \(b'(0)>0\). The following results are proved: if \(a'\in L^ 1(\mathbb{R}_ +)\) then there exists a global solution if the data are sufficiently small (with reference to suitable norms); morover if \(a'\in L^ 1_{loc}(\mathbb{R}_ +)\) and \(0<c\leq b'(x)\leq d\), \(x\in\mathbb{R}\) for \(c,d>0\) constant then there exists a local solution for data of arbitrary size. The method of the proof uses approximating kernels of strongly positive type and energy estimates.
nonlinear Volterra integro- differential equation
scientific article; zbMATH DE number 40684
scientific article