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Generation and characterization of nonlinear semigroups associated to semilinear evolution equations involving “generalized” dissipative operators. (English) Zbl 1146.47037

The authors study the generation of nonlinear semigroups associated to the semilinear problem \[ u'(t)= (A+ B)u(t),\;t> 0;\qquad u(0)= x\in D, \] where \(A\) is the generator of a \(C_0\)-contraction semigroup \(T\) on a Banach space \((X,|\cdot|)\) and \(B: D\to X\) is a nonlinear, continuous operator defined on the closed set \(D\subset X\). They prove that the combination of the subtangential condition
\[ \liminf_{h\downarrow 0}\,(1/h)\,d(T(h)x+ hBx, D)= 0\quad\text{for }x\in D \]
and the semilinear stability condition
\[ \liminf_{h\downarrow 0}\,(1/h)\,(|T(h)(x-y)+h(Bx-By)|-|x-y|)\leq w(|x-y|)\text{ for }x,y\in D, \]
where \(w\) is an increasing uniqueness function, is necessary and sufficient for the generation of a nonlinear semigroup which provides a mild solutions to the problem. If \(B\) is dissipative with respect to \(w\), then the first condition implies the second, and if the second condition is satisfied, then \(A+B\) is strongly dissipative with respect to \(w\). The second condition also guarantees the uniqueness of the mild solution for fixed initial data. The proof is based on a comparison argument together with estimates in terms of solutions of initial value problems for specific ordinary differential equations involving \(w\). Several situations are described in which the conditions are satisfied, and the theory is applied to a concrete example. The paper is methodically related to the work of T.Iwamiya, T.Takahashi and S.Oharu [Lect.Notes Math.1540, 85–102 (1993; Zbl 0819.47081)].

MSC:

47H20 Semigroups of nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations

Citations:

Zbl 0819.47081
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