Abstract
Suppose that x'(t)+Ax(ί) = /(*,*( 0, is a semϋinear parabolic equation, e~Al is bounded and / satisfies the usual continuity condition. If for some 0 1, γ>i, ||/^e- / || 1, whenever ||Λαjt|| 4- ||x|| is small enough, then for small initial data there exist stable global solutions. Moreover, if the space is reflexive then their limit states exist. Some theorems that are useful for obtaining the above bounds and some examples are also presented.
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