| Popis: |
In this paper we consider the semilinear damped wave problem of the form { ( α ( t ) u t ) t − β ( t ) Δ u + γ ( t ) u t + δ ( t ) u = β ( t ) f ( u ) , x ∈ Ω , t > τ , u ( x , t ) = 0 , x ∈ ∂ Ω , t ⩾ τ , u ( x , τ ) = u τ ( x ) , u t ( x , τ ) = v τ ( x ) , x ∈ Ω , where Ω is a bounded smooth domain in R N , N ⩾ 3 , τ ∈ R , f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α , β , γ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space. |
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