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Critical Nonlinear Nonlocal Equations on a Half-Line

Authors:

Elena I Kaikina Leonardo Guardado-Zavala Hector F Ruiz-Paredes Jesus A Mendez Navarro

Institution:

(1) Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia, CP 58089, Michoacán, Mexico;(2) Posgrado en Electrica, Instituto Tecnolόgico de Morelia, CP 58120 Morelia, Michoacán, Mexico

Abstract:

We study nonlinear nonlocal equations on a half-line in the critical case

$\left\{ {\begin{array}{l} {\partial _{{t\,}} u + \,\beta |u|^{\alpha } \,u\, + \,\mathbb{K}u = 0,\,\,\,\,x\, > \,0,\,\,t > \,0,\,} \\ {u(0,x)\, = \,u_{0} \,(x)\,,\,\,\,\,x\, > \,0,} \\ {\partial _{x}^{{j - 1\,}} \,u\,(0,t)\, = \,0,\,j\, = \,1,\,\ldots,\,M} \\ \end{array} } \right.$

where $\beta \in {\mathbf{C}}$ . The linear operator ${\mathbb{K}}$ is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol $K(p) = E_{\alpha}p^{\alpha}$ , the number $M = \frac{\alpha} {2}]$ . The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find the main term of the large time asymptotic representation of solutions in the critical case.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35S15 35B40

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