Oscillation criteria for impulsive parabolic boundary value problem with delay
The authors deal with the following impulsive parabolic system with delay \[ \begin{aligned} u_t= a(t)\Delta_xu+ b(t)\Delta_x u(t- \rho,x)- p(t,x) f(u(t- \sigma,x)), &\quad t\neq t_k,\\ u(t^+_k, x)- u(t^-_k, x)= I(t,x,u), &\quad t= t_k,\;k= 1,2,\dots,\end{aligned}\tag{1} \] where \(\Delta_x\) is the Laplacian in \(\mathbb{R}^n\); \(u= u(t,x)\) for \((t,x)\in \mathbb{R}_+\times\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\partial\Omega\) and \(\mathbb{R}_+= [0,+\infty)\); \(a(t)\), \(b(t)\), \(p(t,x)\) are piecewise continuous functions of their arguments; \(\rho\) and \(\sigma\) are positive constants with \(I: \mathbb{R}_+\times \overline\Omega\times\mathbb{R}\mapsto \mathbb{R}\). For different kind of boundary conditions the authors study oscillation criteria for (1).
scientific article; zbMATH DE number 2082074