On a Nonlinear Hyperbolic Volterra Equation

Source abstract

We study questions of existence, boundedness and asymptotic behavior of the solutions of the initial value problem \[(*)\qquad \begin{array}{*{20}c} {u_t (t,x) - \int_0^t {a (t - s)\sigma (u_x (s,x))_x = f(t,x),\quad 0 < t < \infty ,\quad x \in R.} } \\ {u(0,x) = u_0 (x),\quad x \in R.} \\ \end{array} \] Here $a:R^ + = [0,\infty ) \to R,\sigma :R \to R,f:R^ + \times R \to R,u_0 :R \to R$ are given, sufficiently smooth functions, and the subscripts t and x denote partial derivatives. If $a(t) \equiv 1$, then (*) reduces to a nonlinear wave equation, and it is well known that in this case classical solutions of (*) do not in general exist for all time. However, we show that for a large class of kernels of physical importance equation (*) has global classical solutions for small data. This class of kernels includes all those which are nonconstant, nonnegative, nonincreasing, convex and sufficiently smooth. We also analyze the asymptotic behavior of the solutions.