A quasilinear hyperbolic integro-differential equation related to a nonlinear string

Abstract:

We discuss global existence, boundedness and regularity of solutions to the integrodifferential equation \[ \begin {array}{*{20}{c}} {{u_{t t}} (t,x) + c (t) {u_t}(t,x) - M\;\left ( {\int _{ - \infty }^{ + \infty } {|{u_x}(t,s){|^2}\;ds} } \right )\;{u_{x x}}(t,x) + u (t,x)} \\ { = h (t,x,u (t,x)), \qquad 0 \leq t < \infty ,x \in {\mathbf {R}},} \\ {u (0,x) = {u_0}(x),\quad {u_t}(0,x) = {u_1}(x), \qquad x \in {\mathbf {R}}.} \\ \end {array}\] This type of equation occurs in the study of the nonlinear behavior of elastic strings. We show that if the initial data ${u_0} (x),{u_1} (x)$ is small in a suitable sense, and if the damping coefficient $c (t)$ grows sufficiently fast, then the above equation possesses a globally defined classical solution for forcing terms $h (t,x,u)$ which are sublinear in $u$. In the nonlinearity we require that $M \in {C^1} [0,\infty )$ and, in addition, satisfies $M( \lambda ) \geq {m_0} > 0$ for all $\lambda \geq 0$.