# American Institute of Mathematical Sciences

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September  2009, 2(3): 449-471. doi: 10.3934/dcdss.2009.2.449

## Singularly non-autonomous semilinear parabolic problems with critical exponents

 1 Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Caixa postal 668, 13560-970 São Carlos, São Paulo, Brazil 2 Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos SP, Brazil

Received  September 2008 Revised  December 2008 Published  June 2009

In this work we consider initial value problems of the form

$\frac{dx}{dt} + A(t)x = f(t,x)$
$x(\tau)=x_0,$

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 $t$. We show local well posedness for the case when the nonlinearity $f$ grows critically. Applications to semilinear parabolic equations and strongly damped wave equations are considered.

Citation: Alexandre Nolasco de Carvalho, Marcelo J. D. Nascimento. Singularly non-autonomous semilinear parabolic problems with critical exponents. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 449-471. doi: 10.3934/dcdss.2009.2.449
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