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Abstract Cauchy problems for quasi-linear evolution equations in the sense of Hadamard. (English) Zbl 1058.34079

Let us consider three real Banach spaces \(Y\subset E\subset X\) with dense, continuous inclusions and such that there is an isomorphism \(S:Y\to X\). Moreover, let \(D\) be a closed, bounded subset of \(Y\) and consider the abstract quasilinear Cauchy problem \[ u'(t)= A(t,u(t))u(t),\quad t \in [0,T], \tag{QE} \] in \(X\), where \(\{A(t,w)\); \((t,w)\in [0,T]\times D\}\) is a family of closed linear operators in \(X\) satisfying the following condition: for each \((t,w)\in [0,T]\times D\), the domain of \(A(t,w)\) contains \(Y\) and \(A(t,w)Y\subset E\). This family is assumed to be a strongly continuous family of bounded operators acting from \(Y\) to \(E\).
By a classical solution to (QE) we mean a function in \(C([0,T],D)\cap C^1([0,T],E)\) satisfying (QE) and the initial condition.
To define the notion of well-posedness in the sense of Hadamard, we introduce another Banach space \(X_0\) continuously embedded in \(X\) and such that \(X_0\cap S^{-1}(X_0)\) is dense in \(X_0\), and a subset \(D_0\subset D\) satisfying \(S(D_0)\subset X_0\). The Cauchy problem (QE) is said to be well-posed in the sense of Hadamard if for any \(u(0)\in D_0\) there is a classical solution \(u(t)\) depending continuously on the initial data in the following sense: there is \(M>0\) such that \[ \| u(t)-\hat{u}(t)\| _X\leq M\| u(0)-\hat{u}(0)\| _{X_0}, \quad t\in [0,T]. \] Under several other assumptions, the author proves that (QE) is indeed well-posed in the abovementioned sense. The proofs involve finite difference approximations to (QE) and thus conditions involving solvability and stability of such approximations form an essential part of the assumptions.
The main theorem generalizes Kato’s approach to quasilinear problems and also extends some results on abstract Cauchy problems related to integrated and regularized semigroups. As an application, the author discusses a degenerate equation of Kirchoff’s type.

MSC:

34G20 Nonlinear differential equations in abstract spaces
47D60 \(C\)-semigroups, regularized semigroups
47D62 Integrated semigroups
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