Singularly non-autonomous semilinear parabolic problems withcritical exponents

Alexandre Nolasco de Carvalho; Marcelo J. D. Nascimento

doi:10.3934/dcdss.2009.2.449

Article Contents

2009, Volume 2, Issue 3: 449-471. Doi: 10.3934/dcdss.2009.2.449

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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
2.
Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos SP

Received: August 31, 2008

Revised: November 30, 2008

Published: August 2009

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Abstract

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.

Keywords:

Non-autonomous semilinear parabolic problems,
local existence,
linear evolution process,
critical exponents,
$\epsilon-$regular map.

Mathematics Subject Classification: Primary: 35A07, 35B33, 37B55; Secondary: 35K90, 34G20.

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