tailieunhanh - Báo cáo sinh học: " Research Article Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 946143 29 pages doi 2011 946143 Research Article Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line Martin P. Arciga A Instituto de Matemứticas UNAM Campus Morelia Apartado Postal 61-3 Xangari 58089 Morelia MICH Mexico Correspondence should be addressed to Martin P. Arciga A mparciga@ Received 22 April 2010 Accepted 16 June 2010 Academic Editor Daniel Franco Copyright 2011 Martin P. Arciga A. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We consider the initial-boundary value problem for a nonlinear partial differential equation with module-fractional derivative on a half-line. We study the local and global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. 1. Introduction We study the local and global existence and asymptotic behavior for solutions to the initialboundary value problem ut N u ôx au 0 t 0 x 0 u x 0 u0 x x 0 where N u u ơu Ơ 1 and ôx is the module-fractional derivative operator defined by dx au x t 0 x 2n epx p ail p t dp 1 a 2 2 Boundary Value Problems where u p t is the Laplace transform for u x t with respect to x and d x is the Heaviside function 0 x 1 .0 x 0 x 0. The Cauchy problem for a wide class of nonlinear nonlocal dissipative equations has been studied extensively. In particular the general approach for the study of the large time asymptotics to the Cauchy problem for different nonlinear equations was investigated in the book 1 and the references therein. The boundary value problems are more natural for applications and play an important role in the contemporary mathematical physics. However their mathematical investigations are more complicated even in the