We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in Wasserstein space of functional with competing attractive-repulsive interactions. prove that fully deterministic particle approximations piecewise constant densities introduced [25] starting from general bounded initial converge strongly L 1 to unique weak solution PDE. In particular, result achieved unb...

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or some unbounded domains.

Existence and stability of travelling waves are conveniently studied on unbounded domains , for instance, on the real line. Motivated by numerical and experimental studies, we investigate the eeects of truncating the unbounded domain to a large but bounded domain. It is shown that periodic boundary conditions accurately reproduce the L 2-spectrum of the unbounded domain, whereas separated bound...

When using Laguerre and Hermite spectral methods to numerically solve PDEs in unbounded domains, the number of collocation points assigned inside region interest is often insufficient, parti...

Let Ω be a domain in RN , N ≥ 1, and 2∗ = ∞ if N = 1, 2, 2∗ = 2N N−2 if N > 2, 2 < p < 2 ∗. Consider the semilinear elliptic problem −∆u+ u = |u|p−2u in Ω; u ∈ H 0 (Ω). Let H1 0 (Ω) be the Sobolev space in Ω. The existence, the nonexistence, and the multiplicity of positive solutions are affected by the geometry and the topology of the domain Ω. The existence, the nonexistence, and the multipli...

In this paper we investigate Multi-Objective Integer Linear Programming (MOILP) problems with unbounded feasible region and introduce recession direction for MOILP problems. Then we present necessary and sufficient conditions to have unbounded feasible region and infinite optimal values for objective functions of MOILP problems. Finally we present some examples with unbounded feasible region and fi...

The existence and stability of travelling waves are typically investigated on unbounded domains even though the domains in experiments or numerical simulations are bounded. We are interested in the eeects that the truncation of the unbounded to a large but bounded domain has on the stability properties of a nonlinear wave. These eeects depend upon the boundary conditions that are imposed on the...

For differential inequalities with the $$\infty $$ -Laplacian in principal part, we find conditions for absence of solutions unbounded domains. Examples are given that demonstrate accuracy these conditions.