Oscillation criteria of solutions for a class of impulsive parabolic differential equation (Q699645)

The authors investigate impulsive parabolic equations of the form \( u_t=a(t)\triangle u-p(t,x)u-q(t,x)f(u)\), \(t\neq t_k\), \(u(t_k^+,x)-u(t_k^-,x)=g(t_k,x,u)\) (\(k=1,2,\dots \)), where \(\triangle \) is the Laplace operator in \(G=\mathbb{R}_+\times\Omega \), \(\Omega \) is a bounded domain in \(\mathbb{R}^n\) with a piecewise continuous smooth boundary \(\partial \Omega \). Robin and Dirichlet boundary conditions are taken, i.e. (i) \(\partial u/\partial n+\beta (x)u=h_1(t,x)\) and (ii) \(u=h_2(t,x)\), \((t,x)\in \mathbb{R}_+\times \partial\Omega \) (\(t\neq t_k\), \(k=1,2,\dots \)), \(n\) denotes the unit exterior normal vector to \(\partial\Omega \), the functions \(\beta ,h_1,h_2\) are piecewise continuous. Interesting oscillation criteria are investigated provided some differential inequalities do not have positive solutions.