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Non-homogeneous non-linear damped wave equations in unbounded domains

✍ Scribed by Reinhard Racke

Publisher
John Wiley and Sons
Year
1990
Tongue
English
Weight
431 KB
Volume
13
Category
Article
ISSN
0170-4214

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✦ Synopsis

Abstract

We present a global existence theorem for solutions of u^tt^ − ∂~i~a~ik~ (x)∂~k~u + u~t~ = ƒ(t, x, u, u~t~, ∇u, ∇u~t~, ∇^2^u), u(t = 0) = u^0^, u(=0)=u^1^, u(t, x), t ⪖ 0, __x__ϵΩ.Ω equals ℝ^3^ or Ω is an exterior domain in ℝ^3^ with smoothly bounded star‐shaped complement. In the latter case the boundary condition u|~∂Ω~ = 0 will be studied. The main theorem is obtained for small data (u^0^, u^1^) under certain conditions on the coefficients a~ik~.

The L^p^ ‐ L^q^ decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.

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