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

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 xin mathbb{R}^{+},; t>0, cr u( 0,x) =u_{0}( x) , quad xin 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.

Keywords

• Dissipative nonlinear evolution equation
• Sobolev equation
• large time asymptotic behavior