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Indexed by:期刊论文

Date of Publication:2014-03-01

Journal:SCIENCE CHINA-MATHEMATICS

Included Journals:SCIE

Volume:57

Issue:3

Page Number:555-568

ISSN No.:1674-7283

Key Words:semilinear pseudo-parabolic system; critical Fujita exponent; second critical exponent; global profile

Abstract:This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system u (t) - Delta u -alpha Delta u (t) = v (p) , v (t) -Delta v-alpha Delta v (t) = u (q) with p, q a (c) 3/4 1 and pq > 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the L (p) -L (q) technique to establish some uniform L (m) time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various L (m) time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.