Abstract
AbstractWe present a global existence theorem for solutions of utt − ∂iaik (x)∂ku + ut = ƒ(t, x, u, ut, ∇u, ∇ut, ∇2u), u(t = 0) = u0, u(=0)=u1, u(t, x), t ⪖ 0, xϵΩ.Ω equals ℝ3 or Ω is an exterior domain in ℝ3 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 (u0, u1) under certain conditions on the coefficients aik.The Lp ‐ Lq 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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