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

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     t(=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.