We consider two different abstract Cauchy problems for equations of Sobolev type with operator coefficients in Banach spaces. For the first problem we obtain, under certain conditions on the coefficients, optimal theorems on the existence and non-existence of a solution global in time. In the case when the solution is blown up we obtain upper and lower bounds for the blow-up time. For the second problem we obtain optimal upper and lower bounds for the rate of blow-up of a solution. In each case we give examples in which the operator coefficients have a physical meaning.
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