Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.52 (2020), 1st Issue

It is well known that under appropriate conditions on a double well potential, the associated Hamiltonian system possesses a pair of heteroclinic solutions joining the minima of the potential in addition to in nitely many other homoclinics and heteroclinics that oscillate between these minima. This paper studies the effect on such solutions of replacing the temporal domain, R, by a nite but long time interval.

We review the inde nite sublinear elliptic equation Δu =a(x)u^q in a smooth bounded domain ΩCR^N, with Dirichlet or Neumann homogeneous boundary conditions. Here 0 < q < 1 and a is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in su_x000E_cient and necessary conditions on a and q for the existence of positive solu- tions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched.

In this paper we simulate positive solutions, large solutions and metasolutions of the heterogeneous logistic equation in a disk and an annulus. The numerical methods introduced in this paper are extremely innovative because they make unnecessary determining any previous lifting and solving any decoupled system of ordinary differential equations. Moreover, they can be used to solve non-radially symmetric problems. The models are of a huge interest in Spatial Ecology because they enable us to analyse the effects of the spatial heterogeneity on the evolution of the terrestrial ecosystems. The large solutions and the metasolutions have been computed by the first time in this paper.