A ''sup+ c inf'' inequality for Liouville-type equations by Bartolucci D.

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By Bartolucci D.

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3. 16) where f E L'(0, T ; X )and x E X. 16). 15) if it is continuously differentiable on [0, TI, u ( t ) E D ( A ) for all t E [O, TI, and it satisfies Eq. 16) on [0, TI. 16) has at most one classical solution u ( t) . Indeed, if y is a solution to dY dt - = Ay in then, for 0 I s I t < m, we have Hence e A ( ' - s ) y ( s= ) 0. [O,m), y ( 0 ) = 0, 1. 161, we have the following well-known theorem whose proof will be omitted. 3. Assume that A is the infinitesimal generator of a C,-semigroup e A ron X and x E D ( A ) .

Hence, S(z) is analytic in A = { z ; larg ZI < arctan l/C}. 4. Linear Differential Equations in Banach Spaces 25 where r is a piecewise smooth curve in A of the form encountered before. 12). 3. 16) where f E L'(0, T ; X )and x E X. 16). 15) if it is continuously differentiable on [0, TI, u ( t ) E D ( A ) for all t E [O, TI, and it satisfies Eq. 16) on [0, TI. 16) has at most one classical solution u ( t) . Indeed, if y is a solution to dY dt - = Ay in then, for 0 I s I t < m, we have Hence e A ( ' - s ) y ( s= ) 0.

4. Let ( x , ) c X a n d {y,} c X * be such that x , -+ 0 and lly,ll + as n -+ m. Let B(0,r) be the closed ball { x ; llxll I r}. 38) Prooj Suppose that the lemma is false. Then there exists r > 0 such that for every u E B(0, r ) there exists C , > -m such that V n E N. , K. Yosida [l],p. 111, we infer that there is k , such that int(u E B ( 0 , r ) ;( x , In other words, there are ( u ; IIu - uoll I E} E > 0, k , c (u E > - k , V n } z 0. - u,y,) N, and u, B(0,r);( x , - E E B(0, r ) such that u,y,) > -k, V n } .

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