On the unboundedness of the ratio of species and resources for the diffusive logistic equation

Abstract

proposed an optimization problem to consider the supremum of the ratio of the L 1 norms of species and resources by varying the diffusion rates and the profiles of resources, and moreover, he gave a conjecture that the supremum is 3 in the one-dimensional case. In [1], Bai, He and Li proved the validity of this conjecture. The present paper shows that the supremum is infinity in a case when the habitat is a multi-dimensional ball. Our proof is based on the sub-super solution method. A key idea of the proof is to construct an L 1 unbounded sequence of sub-solutions.

The bang-bang property in some parabolic bilinear optimal control problems \emph{via} two-scale asymptotic expansions

“…for some time horizon T > 0. This problem is the parabolic counterpart of a related elliptic optimisation problem that was intensively studied in the past few years, see section 1.6 and [20,23,24,28,30,31,35,39,40,43,44]. In the elliptic case, the bang-bang property for optimisers was, in particular, a question that drew a lot of attention [35,40,44] and was only recently settled in [39].…”

Effects of nonlocal dispersal strategies and heterogeneous environment on total population

“…This conjecture is confirmed in [3]. However, for higher dimensional case, i.e., n ≥ 2, it is proved in [16] that the supremum of E(m) is unbounded. Some further studies related to this question can be found in [24,25].…”

Impact of Regional Difference in Recovery Rate on the Total Population of Infected for a Diffusive SIS Model

“…The proof is an application of our result [20] on the profile of the positive stationary solution to a diffusive logistic equation. In [20], the authors proved that some spatial concentration setting of a resource function in the diffusive logistic equation makes the L 1 norm of the positive stationary solution become as large as possible. It will be shown that, in the case of d S = d I , (3) can be reduced to a single equation, which is similar to the stationary diffusive logistic equation.…”

“…By the setting of m(x) := β − γ(x) and v(x) := β w I(x), the function v(x) satisfies the diffusive logistic Equation ( 9). Subsequently, we can apply the results in [20] to v(x) (I(x)). Here, we define a positive constant…”