The formulation of set differential equations has an intrinsic disadvantage that the diameter of the solution is nondecreasing as time increases and therefore the behavior of solutions, in some cases, do not match with the solutions of ordinary differential equations from which set differential equations can be generated. In this paper an approach is provided to remove the disadvantage.

A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation utt-ux, + g(u) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g(u)/u+m as Iu(+w. The proof uses a modified form (PS), of the Palais-Smale condition (PS). Let g : R-+R be a continuous nondecreasing fu...

In this paper, we prove that the first eigenvalues of −∆+ cR (c ≥ 1 4 ) are nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized Ricci flow for the cases c = 1/4 and r ≤ 0. 1. First eigenvalue of −∆+ cR Let M be a closed Riemannian manifold, and (M,g(t)) be a smooth solution to the Ricci flow equation ∂ ∂t gij = −2Rij on 0 ≤ t < T . In [Cao07], we prove that a...

In this paper, we propose a discretization for the compressible Stokes problem with an equation of state of the form p = φ(ρ) (where p stands for the pressure, ρ for the density and φ is a superlinear nondecreasing function from R to R). This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element technique. Th...

We study the existence and uniqueness of a maximal solution of equation ut − ∆u + f(u) = 0 in Ω× (0,∞), where Ω is a domain with a non-empty compact boundary, which satisfies u = g on ∂Ω × (0,∞), assuming that g and f are given continuous functions and f is also convex, nondecreasing, f(0) = 0 and verifies Keller-Osserman condition. We show that if the boundary of Ω satisfies the parabolic Wien...

We are concerned with singular elliptic equations of the form −∆u = p(x)(g(u) + f (u) + |∇u| a) in R N (N ≥ 3), where p is a positive weight and 0 < a < 1. Under the hypothesis that f is a nondecreasing function with sublinear growth and g is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on...