Electron. J. Diff. Eqns., Vol. 2007(2007), No. 109, pp. 1-11.

Initial-boundary value problems for nonlinear pseudoparabolic equations in a critical case

Elena I. Kaikina

Abstract:
We study nonlinear pseudoparabolic equations, on the half-line in a critical case,
$$\displaylines{ \partial _{t}( u-u_{xx}) -\alpha u_{xx}=\lambda |u| u,\quad x\in \mathbb{R}^{+},\; t>0, \cr u( 0,x) =u_{0}( x) , \quad x\in \mathbb{R}^{+}, \cr u(t,0)=0, }$$
where $\alpha >0$, $\lambda \in \mathbb{R}$. The aim of this paper is to prove the existence of global solutions to the initial-boundary value problem and to find the main term of the asymptotic representation of solutions.

Submitted March 22, 2007. Published August 7, 2007.
Math Subject Classifications: 35Q35
Key Words: Dissipative nonlinear evolution equation; Sobolev equation; large time asymptotic behavior.

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Elena I. Kaikina
Instituto de Matemáticas
Universidad Nacional Aut&ocute;noma de México
Campus Morelia, AP 61-3 (Xangari)
Morelia CP 58089, Michoacán, Mexico
email: ekaikina@matmor.unam.mx

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