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Time-dependent nonlinear evolution equations. (English) Zbl 1041.34049

In this paper, the following evolution inclusion is investigated: \[ \frac{du}{dt}\in A(t)u,\quad u(0)=u_0, \] where \(A(t):D(A(t))\subset X\rightarrow X\) is a nonlinear multivalued operator, \(X\) a Banach space and \(0<t<T\). The main aim of the author is to provide conditions under which (1) can be applied to parabolic boundary value problems with time dependent boundary conditions. Moreover, these conditions have to imply that a generalized solution of (1) is also a strong one.
To this end \(A(t)\) is supposed to satisfy a dissipativity, a compactness and an appropriate range condition. Additionally, some continuity of \(J_\lambda(t)x=(I-\lambda A(t))^{-1}\) is needed.
The obtained results generalize a result by M. G. Crandall and A. Pazy [Isr. J. Math. 11, 57–94 (1972; Zbl 0249.34049)], where the domains of the \(A(t)\)’s were supposed to be independent of \(t\).

MSC:

34G25 Evolution inclusions
47B44 Linear accretive operators, dissipative operators, etc.
47H20 Semigroups of nonlinear operators
47J35 Nonlinear evolution equations

Citations:

Zbl 0249.34049
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