REMARKS ON BLOW-UP AND NONEXISTENCE THEOREMS FOR NONLINEAR EVOLUTION EQUATIONS

OVER the last 20 years a large literature has developed concerning evolution equations which for certain initial data possess solutions that do not exist for all time. The bulk of this literature relates to problems arising from partial differential equations. To establish nonexistence it is customary to argue by contradiction. One supposes that for given UQ and t0 a solution u(t) with u(ro)= u0 exists for all times t^t0; typically u takes values in some Banach space X and we will assume that this is the case. A function p : X —» R is then constructed, and by use of differential inequalities it is shown that lim p(u(()) = ° for some tle(t0, °°). This usually

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