On the dynamics of a~nonlinear multidimensional oscillator with memory (Q1586982)

Assume that \(x(t)=(x_1(t),\dots,x_N(t))\) is a~real-valued vector, \(V=V(x)\) is a~real-valued function, \(\nabla V(x)=({\partial V\over\partial x_1},\dots, {\partial V\over\partial x_N})\), \( F=F(x,y)\) is a~real-vector-valued function of real-valued vector arguments, and \( K=K(t)=(K_{ij}(t)) \), \( i,j=1,\dots ,N \), is a real matrix-valued function of the time~\(t\). The author studies the Cauchy problem for the system of nonlinear integro-differential equations \[ \begin{gathered} \partial_t^2 x(t)+ \nabla V(x(t))+F\bigl(x(t), \partial_t x(t)\bigr)-\int_0^t K(t-t_1) x(t_1) dt_1=f(t), \\ x(0)=x_0,\quad \partial_t x(0)=y_0. \end{gathered} \] The author suggests conditions for this Cauchy problem to have a~unique solution on the half-axis \(t\geq 0\). Moreover, the author gives some conditions for which the Cauchy problem has a~unique solution with \(|x(t) |\) and \(|\partial_t x(t) |\) bounded and studies the behavior of the solution as \(t\to +\infty\). Note that the above system generalizes many classical mechanical systems with memory.