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Trend to spatial homogeneity for solutions to semilinear damped wave equations

Synopsis The semilinear damped wave equations subject to homogeneous Neumann boundary conditions, admit spatially homogeneous solutions (i.e. u(x, t) = u(t)). In order that every solution tends to a spatially homogeneous one, we look for conditions on the coefficients a and d, and on the Lipschitz constant of f with respect to u.