Some non-local elliptic problems. Applications to population dynamics and cancer stem-cells models
Antonio Suárez Fernández (Universidad de Sevilla, Espanha)
In the classical local partial differential equations, the relation between the unknown variable and its derivatives is local in space, that is, in the equation these functions are all taken in the same point of the domain.
However, for example in ecological context, there is no real justification for assuming that these interactions are local; in fact, there are many examples that show that they are non-local. Including non-local terms in partial differential equations brings some important difficulties. Among others, the problem has not a variational structure not verifies, in general, the maximum principle. We present some theoretical results (existence, uniqueness or multiplicity of positive solutions) of some non-local problems arising from population dynamics and cancer stem-cells. Mainly, we employ sub-supersolutions, bifurcations and topological methods to obtain our results.