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'... . 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 = ...'

. 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...

'... We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based ...'

We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based on comparison with global solutions. For a critical power non-linearity we obtain a two-parameter family of radially symmetric stationary solutions. By extending

'... We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X , u tt = f 0 (u); t > 0; u(0) = u 0 ; u t (0) = u 1 ; where f : X ! R is a C¹-function. Several applications to the second and higher-order hyperbolic equations with local and nonloca ...'

We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X , u tt = f 0 (u); t > 0; u(0) = u 0 ; u t (0) = u 1 ; where f : X ! R is a C¹-function. Several applications to the second and higher-order hyperbolic equations with local and nonlocal nonlinearities are presented. We also describe an approach to Kato's and John's critical exponents for the semilinear equations u t = u+b(x; t)juj p , p > 1, which are responsible for phenomena of stability, unstability, blow-up and asymptotic behaviour. We construct countable spectra of different asymptotic patterns of self-similar and non self-similar types for global and blow-up solutions for the autonomous equation u tt = u + juj p 1 u in different parameter ranges.

'... We prove L estimates on the blow-up behaviour of solutions of a 2m-th order semilinear parabolic equation u t = ( ) u + q(u); x 2 R ; t > 0; m > 1; with a general even function q(u) 0 with a superlinear growth for juj 1. Our comparison approach and estimates apply to general int ...'

We prove L estimates on the blow-up behaviour of solutions of a 2m-th order semilinear parabolic equation u t = ( ) u + q(u); x 2 R ; t > 0; m > 1; with a general even function q(u) 0 with a superlinear growth for juj 1. Our comparison approach and estimates apply to general integral evolution equations. We also study the following problem: nd a continuous function q(u) with a superlinear growth as u !1 such that the parabolic equation exhibits regional blow-up in a domain of nite non-zero measure. We show that such a regional blow-up can occur for q(u) = uj ln jujj . We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t ! T is described by the self-similar solution U (x; t) = expf(T t) (x)g; : R ! C ; of the complex Hamilton-Jacobi equation U t = ( 1) 1 2m (rU rU)