Asymptotic Analysis

We give an extended Pizzetti asymptotic formula, namely, we show that if ϕ is smooth in an open set that contains the origin in R n and if p is a harmonic homogeneous polynomial of degree k, then (0.1) ∫ S ϕ ( ε ω ) p ( ω ) d σ ( ω ) ∼ C ∑ m = 0 ∞ Δ m p ( ∇ ) ϕ ( 0 ) W n , k , m ε k + 2 m , as ε → 0 + for some constants C and W n , k , m that are given in the text. We also show that the formula never holds, for all ϕ, if p is not harmonic.

We consider two procedures, one algebraic and the other analytic, to find the part of an object – a formal power series or a smooth function – that is a radial multiple of a given polynomial and show that the two constructions yield the same results for harmonic polynomials but not otherwise.

We also consider mean value type results for solutions of partial differential equations, including a version of Morera’s theorem that applies to locally integrable functions.

In this paper, we describe the asymptotic behavior of the scattering matrix S associated with the 2 × 2 matrix Schrödinger operator P = − h 2 d 2 d x 2 I 2 + V ( x ) + h R ( x , h D x ) on L 2 ( R ) ⊕ L 2 ( R ) . We study the situation where the diagonal terms of V cross on the real axis. In particular it is proven that, due to the real crossing, the off-diagonal terms of S are no longer exponentially small with respect to the semi-classical parameter h.

We study the homogeneous Dirichlet problem for the class of nonlinear parabolic equations with variable nonlinearity u t − div ( D ( x ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , t , u ) − A ( x ) | u | q ( x ) − 2 u in the cylinder Ω × ( 0 , T ) with given nonnegative weights D ( x ) , A ( x ) , measurable bounded exponents p ( x ) ∈ [ p − , p + ] , q ( x ) ∈ [ q − , q + ] and a globally Lipschitz function f ( x , t , u ) . Sufficient conditions of existence and uniqueness of weak and strong solutions are derived. We find conditions on the exponents p ( x ) , q ( x ) which guarantee that the associated semigroup has a compact global attractor in L 2 ( Ω ) . It is shown that in case the exponents p ( x ) and q ( x ) do not meet the sufficient conditions of existence of a nontrivial global attractor and ‖ u ( 0 ) ‖ L 2 ( Ω ) is sufficiently small, then every solution with bounded ‖ u ( t ) ‖ L 2 ( Ω ) 2 either vanishes in a finite time, or decays exponentially as t → ∞ .