Death Row: An Existence of Non-Existence

Existence and non-existence of solutions for an elliptic system

We study the existence of positive solutions for a system of two elliptic equations of the form  −∆u = a1(x)F1 (x, u, v) in Ω −∆v = a2(x)F2 (x, u, v) in Ω u = v = 0 on ∂Ω where Ω ⊂ RN (N ≥ 2) is a bounded domain in RN with a smooth boundary ∂Ω or Ω = RN (N ≥ 3). A non-existence result is obtained for radially symmetric solutions. Our proofs are based primarily on the sub and super-solution met...

Existence and non-existence of skew branes

A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any two distinct points are not parallel. We show that if an oriented closed manifold has a non-zero Euler characteristic χ then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than χ/4. We give a version of this result for immersed surf...

Existence of an Erythropoitic labile iron pool

The (non-)existence of perfect codes in Lucas cubes

A Fibonacci string of length $n$ is a binary string $b = b_1b_2ldots b_n$ in which for every $1 leq i < n$, $b_icdot b_{i+1} = 0$. In other words, a Fibonacci string is a binary string without 11 as a substring. Similarly, a Lucas string is a Fibonacci string $b_1b_2ldots b_n$ that $b_1cdot b_n = 0$. For a natural number $ngeq1$, a Fibonacci cube of dimension $n$ is denoted by $Gamma_n$ and i...

Existence and Non-existence of Fisher-KPP Transition Fronts

We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion transition front exists. A weaker localized...