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Non‐homogeneous non‐linear damped wave equations in unbounded domains
Author(s) -
Racke Reinhard
Publication year - 1990
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670130604
Subject(s) - mathematics , ansatz , eigenfunction , bounded function , domain (mathematical analysis) , mathematical analysis , homogeneous , complement (music) , mathematical physics , wave equation , boundary value problem , a priori estimate , pure mathematics , combinatorics , physics , eigenvalues and eigenvectors , quantum mechanics , biochemistry , chemistry , complementation , gene , phenotype
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.