Oscillations of solutions of vector differential equations of parabolic type with functional arguments
This paper is devoted to the oscillatory properties of solutions of the vector parabolic differential equation with functional arguments \[ \begin{multlined} \frac{\partial} {\partial t} \left(U(x,t)+ \sum^\ell_{i=1} h_i(t)U \bigl(x,\rho_i(t)\bigr)\right) -a(t)\Delta U(x,t)-\sum^k_{i=1} b_i(t)\Delta U\bigl(x,\tau_i(t)\bigr)+\\ +\sum^m_{i=1} c_i \biggl(x,t,U \bigl(x, \sigma_i(t)\bigr) \biggr)U\bigl(x,\sigma_i(t)\bigr)=F(x,t),\;(x,t)\in \Omega =G\times(0,\infty)\end{multlined}\tag{1} \] where \(G\) is a bounded domain in \(\mathbb{R}^n\) with piecewise smooth boundary \(\partial G\) and \(h_i,b_i,a, \rho_i,\tau_i, \sigma_i\), \(c_{ix}\) are given functions. Under some natural assumptions on the data of (1), the authors reduce the multi-dimensional oscillation problem (1) to one-dimensional oscillation problems for scalar functional differential inequalities. As a result they present sufficient conditions for every solution of boundary value problems for (1) with functional arguments to be \(H\)-oscillatory in a cylindrical domain.
parabolic differential equation with functional arguments
oscillation problems for scalar functional differential inequalities
scientific article; zbMATH DE number 1882243