## 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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