Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary (Q713343)

The authors considered the solutions for a parabolic problem with nonlinear boundary conditions and delay in the boundary of the domain. By constructing a nonlinear reaction-diffusion problem with delay in the interior of the domain, with the reaction-diffusion term concentrated in a neighborhood of the domain. The neighborhood depends on a parameter \(\varepsilon\) which shrinks to the boundary as \(\varepsilon\to 0\). Assuming that the nonlinearity \(f\) has a linear upper bound, they showed that the weak solutions of the linear reaction-diffusion problem with delay in the interior of the domain are globally defined and bounded. By comparison, they obtained the global existence of the weak solutions of the nonlinear reaction-diffusion problem with delay in the interior of the domain. Finally, a convergence theorem was obtained as \(\varepsilon\to 0\) which guarantees the existence of solutions of the original problem for the delay in the boundary.