0 Growth Estimates on Positive Solutions of the Equation ∆ u + Ku n + 2 n − 2 = 0 in IR n Man Chun L EUNG

A pr 1 99 9 Asymptotic Behavior of Positive Solutions of the Equation ∆ u + Ku n + 2 n − 2 = 0 in IR n and Positive Scalar

The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds

Abstract. We study the equation ∆gu− n−2 4(n−1)R(g)u+Ku p = 0 (1+ ζ ≤ p ≤ n+2 n−2 ) on locally conformally flat compact manifolds (M, g). We prove the following: (i) When the scalar curvature R(g) > 0 and the dimension n ≥ 4, under suitable conditions on K, all positive solutions u have uniform upper and lower bounds; (ii) When the scalar curvature R(g) ≡ 0 and n ≥ 5, under suitable conditions ...

Existence and Uniqueness of the Entropy Solution to a Nonlinear Hyperbolic Equation

This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: u t + div(vf (u)) = 0 in IR N 0; T ], with initial data u(:; 0) = u 0 (:) in IR N. where u 0 2 L 1 (IR N) is a given function, v is a divergence-free bounded function of class C 1 from IR N 0; T ] to IR N , and f is a function of class C 1 from IR to...

Some Compactness Results Related to Scalar Curvature Deformation

Motivated by the prescribing scalar curvature problem, we study the equation ∆gu+Ku p = 0 (1 + ζ ≤ p ≤ n+2 n−2 ) on locally conformally flat manifolds (M, g) with R(g) = 0. We prove that when K satisfies certain conditions and the dimension of M is 3 or 4, any solution u of this equation with bounded energy has uniform upper and lower bounds. Similar techniques can also be applied to prove that...

Decay Estimates for Fourth Order Wave Equations

may be thought of as a nonlinear beam equation. In this paper we obtain both L-L estimates and space-time integrability estimates on solutions to the linear equation. We also use these estimates to study the local existence and asymptotic behavior of solutions to the nonlinear equation, for nonlinear terms which grow like a certain power of u. The main L-L estimate (Theorem 2.1) states that sol...