Regularity of solutions on the global attractor for a semilinear damped wave equation

Abstract

We consider attractors A η , η ∈ [ 0 , 1 ] , corresponding to a singularly perturbed damped wave equation u t t + 2 η A 1 2 u t + a u t + A u = f ( u ) in H 0 1 ( Ω ) × L 2 ( Ω ) , where Ω is a bounded smooth domain in R 3 . For dissipative nonlinearity f ∈ C 2 ( R , R ) satisfying | f ″ ( s ) | ⩽ c ( 1 + | s | ) with some c > 0 , we prove that the family of attractors { A η , η ⩾ 0 } is upper semicontinuous at η = 0 in H 1 + s ( Ω ) × H s ( Ω ) for any s ∈ ( 0 , 1 ) . For dissipative f ∈ C 3 ( R , R ) satisfying lim | s | → ∞ f ″ ( s ) s = 0 we prove that the attractor A 0 for the damped wave equation u t t + a u t + A u = f ( u ) (case η = 0 ) is bounded in H 4 ( Ω ) × H 3 ( Ω ) and thus is compact in the Hölder spaces C 2 + μ ( Ω ¯ ) × C 1 + μ ( Ω ¯ ) for every μ ∈ ( 0 , 1 2 ) . As a consequence of the uniform bounds we obtain that the family of attractors { A η , η ∈ [ 0 , 1 ] } is upper and lower semicontinuous in C 2 + μ ( Ω ¯ ) × C 1 + μ ( Ω ¯ ) for every μ ∈ ( 0 , 1 2 ) .