Study of nonlinear PDE with power nonlinearities

Abstract

Let u ′ + A ⁢ u = h ⁢ ( u , t ) + f ⁢ ( x , t ) , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) , where u ∈ H , u ′ := u t := d ⁢ u d ⁢ t , H is a Hilbert space, ∥ h ⁢ ( u , t ) ∥ ≤ a ⁢ ∥ u ∥ p ⁢ ( 1 + t ) - b , ( A ⁢ u , u ) ≥ γ ⁢ ( t ) ⁢ ( u , u ) , γ ⁢ ( t ) = c ⁢ ( 1 + t ) - c 1 . Here p , a , b , c , c 1 are positive constants. Sufficient conditions are given for the above problem to have a solution bounded or going to zero as t → ∞ .