. As a basic example, we establish that in the Cauchy problem for the 2m-th order semilinear parabolic equation u t = ( ) m u + juj p ; x 2 R N ; t > 0; u(x; 0) = u 0 (x); x 2 R N ; where m > 1, p > 1, with bounded integrable initial data u 0 , the critical Fujita exponent is pF = 1 + 2m=N , so that for p > pF there exists a class of small global solutions and for p 2 (1; pF ] blow-up can occur for arbitrarily small initial data. The analysis of the asymptotics of both classes of global and blow-up solutions is based on comparison with similarity solutions of the majorizing order-preserving equation, which is shown to exist for any m > 1. Generalizations of this idea to dierential and pseudodierential evolution equations and relations to positivity sets for higher-order equations are discussed. 1. Introduction: majorizing order-preserving operators and equations This paper deals with a class of higher-order semilinear parabolic dierential and nonlinear integral evolution...
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Galaktionov et al. (2002) studied this question.