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Estimates for a class of oscillatory integrals and decay rates for wave-type equations
 

Summary: Estimates for a class of oscillatory integrals and
decay rates for wave-type equations
JinMyong Kim, Anton Arnold and Xiaohua Yao
Abstract. In this paper we first establish global pointwise time-space esti-
mates for a class of oscillatory integrals. Then, we use them to establish
Lp
- Lq
estimates for a class of higher order wave-type equations of the form
ttu + P(Dx)u = 0, where the symbol P() is a real non-degenerate elliptic
polynomial of Rn
.
Mathematics Subject Classification (2000). 42B25; 42A45; 35Q53; 35Q35.
Keywords. Oscillatory integral, higher-order wave equation, fundamental so-
lution estimate.
1. Introduction
It is well known that the solution u(t, x) of the Cauchy problem for the wave
equation:
ttu(t, x) - u(t, x) = 0, (t, x) R × Rn
u(0, x) = u0(x), tu(0, x) = u1(x), x Rn
has the following form:

  

Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien

 

Collections: Mathematics