Abstract
We prove that any bounded solution (u, u 1) ofu u +du t −Δu+f(u)=0,u=u(x, t), x∈ℝN,N⩾3, converges to a fixed stationary state provided its initial energy is appropriately small. The theory of concentrated compactness is used in combination with some recent results concerning the uniqueness of the so-called ground-state solution of the corresponding stationary problem.
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Feireisl, E. Long-time behavior and convergence for semilinear wave equations on ℝN . J Dyn Diff Equat 9, 133–155 (1997). https://doi.org/10.1007/BF02219055
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DOI: https://doi.org/10.1007/BF02219055