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Singularly non-autonomous semilinear parabolic problems with critical exponents. (English) Zbl 1180.35320

The authors consider singularly non-autonomous semilinear abstract parabolic problems of the form \[ \left\{ \begin{aligned} \frac{dx}{dt}+A(t)x=f(t,x), & t>0\\ x(\tau)=x_0\in D,& \end{aligned} \right. \] in a Banach space \(X\) where \(A(t):D\subset X\to X\) is a linear, closed and unbounded operator which is sectorial for each \(t\), \(f:\mathbb{R}\times D\to X\) is critical (has the same order as \(A(t)\)). The authors show local well posedness for the case when the nonlinearity \(f\) grows critically. Applications to semilinear parabolic equations and strongly damped wave equations are given.

MSC:

35K90 Abstract parabolic equations
35B33 Critical exponents in context of PDEs
37B55 Topological dynamics of nonautonomous systems
35K58 Semilinear parabolic equations
34G20 Nonlinear differential equations in abstract spaces
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