Estimates of solutions of impulsive parabolic equations under Neumann boundary condition
The paper deals with the initial-boundary value problem (IBVP) for the impulsive nonlinear parabolic equations \[ u_t(t,x) = \rho \Delta u(t,x) + f(t,x,u(t,x)), \quad t \neq t_k, x \in \Omega, \] \[ u(0,x) = u_0(x), \quad x \in \overline{\Omega}, \] \[ \frac{\partial u(t,x)}{\partial \nu} = 0, \quad t \neq t_k, x \in \partial \Omega, \] \[ u(t_k,x) = u(t_k^-,x) + g(t_k,x,u(t_k^-,x)), \quad x \in \overline{\Omega}, \;k = 1,2,\dots , \] where \(t_1 < t_2 < \dots < t_k < \dots \) are given numbers such that \(\lim_{k \to \infty} t_k = \infty\). The authors obtain estimates of the solutions of the IBVP (1)--(4) by the solutions of adequate impulsive ordinary differential equations. An interesting application of the results is given for a real problem from the population dynamics.
scientific article; zbMATH DE number 1970831