GLOBAL EXISTENCE FOR MILD SOLUTIONS TO SEMILINEAR EVOLUTION EQUATIONS UNDER GENERALIZED DISSIPATIVITY CONDITIONS

Abstract

Let X be a real Banach space, let A : D(A) subset X-> X be the generator of a (C0)-contraction semigroup on X and let B: D  subset [0, T) X-> X be a continuous operator. Under a combination of Pavel's subtangential condition, a semilinear stability condition defined in terms of a uniqueness function w: [0,T) X R -> R and suitable connectedness and closedness asumptions on the domain D of the operator B, we prove the global existence of the mild solution to the equation u0 = Au + B(t, u). In our setting, no dissipativity property is assumed for the operator B.