The porous medium equation with a concentrated nonlinear source

Abstract

In this article, we consider the Cauchy problem for the porous medium equations with a concentrated nonlinear source. We establish the existence of generalized solutions for the problem based on some iteration technique and approximate process.

A diffusive logistic equation with concentrated and nonlocal sources

“…It may happen, for example, if there exists in B a big concentration of predators or the environment has abundant resources within the ball B . Equations with concentrated sources were considered in , for instance.…”

“…On the other hand, the concentrated term d @ B .x/ @ juj 1 u means that the growth rate can be suddenly changed at the rate d inside the ball B. It may happen, for example, if there exists in B a big concentration of predators or the environment has abundant resources within the ball B. Equations with concentrated sources were considered in [8][9][10][11][12], for instance.…”