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Extra resources for Nonlinear elliptic and parabolic equations
Example text
O,. ] + (-- t)'[A if, x, D%), Dk/I+ Sf.
Given a contractive semigroup Tt(u), what is its generator like? , R(I - A) = H (I is the identity o p e r a t o r ) . Our a i m is to e s t a b l i s h a nonlinear analogue of this t h e o r e m . To this end we f i r s t introduce the concept of a d i s s i p a t i v e nonlinear o p e r a t o r . 4. An o p e r a t o r A(u) is called d i s s i p a t i v e if for any e l e m e n t s u E D(A) and v ~ D(A) the following inequality holds: Re
Dt-' f[r = f t-x (7). 1. L e t the conditions I - I V be satisfied. 3) u(t, x) ~ W(f). 514 We c a r r y out the p r o o f by the F a e d o - G a l e r k i n - H o p f method taking account of the monotonicity p r o p e r ties. Conceptually we a r e a l r e a d y v e r y f a m i l i a r with this method, and we shall t h e r e f o r e be brief. ok L e t vl(x), v2(x), . . be a c o m p l e t e s y s t e m in the s p a c e W (a, b). 2) for the function zn(t, x) give Znl F = 0 , . . , D r - l z n I F = 0, DZrznlF = 0 .