Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains

We consider blow-up results for a system of inhomogeneous wave inequalities in exterior domains. We will handle three type boundary conditions: Dirichlet type, Neumann type and mixed boundary conditions. We use a unified approach to show the optimal criteria of Fujita type for each case. Our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation. We provide many new results and close some open questions

Positive solutions to a class of random operator equations and applications to stochastic integral equations

We study the existence of random positive solutions to a random operator equation on ordered Polish spaces. We apply the results obtained in this paper to study the existence of random positive solutions to some classes of stochastic integral equations

Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

We provide sufficient conditions for the nonexistence of global positive solutions to the nonlocal evolution equation utt(x,t)=(J ∗ u-u)(x,t)+up(x,t), (x,t)∈RN×(0,∞),(u(x,0),ut(x,0))=(u0(x),u1(x)), x∈RN, where J:RN→R+, p>1, and (u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). Next, we deal with global nonexistence for certain nonlocal evolution systems. Our method of proof is based on a duality argument

Existence of positive solutions to an arbitrary order fractional differential equation via a mixed monotone operator method

In this paper, using a mixed monotone operator method, we study the existence and uniqueness of positive solutions to a nonlinear arbitrary order fractional differential equation. An example is provided to illustrate the main result

Construction of compact constant mean curvature hypersurfaces with topology

In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean curvature surface homologous to a subset of the interior boundary components. This bound allows us to find a maximizer for the constant mean curvature of a surface homologous to the inner boundary. With this maximizer at hand, we can construct an increasing family of sets with boundaries of increasing constant mean curvature. We interpret this familiy as a weak version of a CMC foliation