Evolution inclusions with Clarke subdifferential type in Hilbert space

In this paper we mainly consider the following evolution inclusion with Clarke subdifferential: (∗) { − x ̇ ( t ) ∈ ∂ c φ ( x ( t ) ) + G ( t , x ( t ) ) x ( 0 ) = x 0 where ∂ c is the Clarke subdifferential and t ∈ I = [ 0 , T ] is a nonempty, bounded closed interval in R + = [ 0 , + ∞ ) ; H is a separable Hilbert space; x 0 ∈ D ( ∂ c φ ) = { x ∈ H : ∂ c φ ( x ) ≠ ϕ } . First, we introduce some hypotheses on φ , under which we prove that there is a unique solution of (∗) when G is a single function. Based on this result, we obtain an existence theorem for G being a u.s.c. or l.s.c multifunction. Then the existence of an extremal solution of (∗) and the relaxation theorem are studied. In the end, we present two examples of PDE to illustrate the application of our results.