Nonnegative solutions of parabolic operators with low-order terms

Uniqueness/nonuniqueness for Nonnegative Solutions of a Class of Second-order Parabolic Equations

We investigate uniqueness and nonuniqueness for nonnegative solutions of the equation ut = Lu+ V u− γu p in R × (0,∞); u(x, 0) = f(x), x ∈ R; u ≥ 0, where γ > 0, p > 1, γ, V ∈ C(R), 0 ≤ f ∈ C(R) and L = Pn i,j=1 ai,j(x) ∂ ∂xi∂xj + Pn i=1 bi(x) ∂ ∂xi with ai,j , bi ∈ C (R).

Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems

Oscillatory Integral Operators with Low–order Degeneracies

We prove sharp L estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation C ⊂ T ∗ΩL × T ∗ΩR projects to T ∗ΩL and to T ∗ΩR with corank one singularities of type ≤ 2. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line ...

On the Uniqueness of Nonnegative Solutions of Differential Inequalities with Gradient Terms on Riemannian Manifolds

We investigate the uniqueness of nonnegative solutions to the following differential inequality div(A(x)|∇u|∇u) + V (x)u1 |∇u|2 ≤ 0, (1) on a noncompact complete Riemannian manifold, where A, V are positive measurable functions, m > 1, and σ1, σ2 ≥ 0 are parameters such that σ1 + σ2 > m− 1. Our purpose is to establish the uniqueness of nonnegative solution to (1) via very natural geometric assu...

MULTIPLE NONNEGATIVE SOLUTIONS FOR BVPs OF FOURTH-ORDER DIFFERENCE EQUATIONS

First, existence criteria for at least three nonnegative solutions to the following boundary value problem of fourth-order difference equation Δ4x(t− 2) = a(t) f (x(t)), t ∈ [2,T], x(0)= x(T +2)=0, Δ2x(0)=Δ2x(T)=0 are established by using the well-known LeggettWilliams fixed point theorem, and then, for arbitrary positive integerm, existence results for at least 2m− 1 nonnegative solutions are ...