Large time asymptotics for the Ott-Sudan-Ostrovskiy type equations on a segment. (English) Zbl 1212.35213

Kaikina, Elena I.

Differ. Integral Equ. 20, No. 12, 1363-1388 (2007).

Summary: We study the initial-boundary value problems for the nonlinear nonlocal equation on a segment \((0,a)\), \(u_t+\lambda | u| u+C_1\int _0^x u_{ss}(s,t)/{\sqrt {x-s}}\,ds=0\), \(t>0\), \(u(x,0)=u_0(x)\), \(u(a,t)=h_1(t)\), \(u_x(0,t)=h_2(t)\), \(t>0\), where \(\lambda \in \mathbb R\) and the constant \(C_1\) is chosen by the condition of the dissipation, such that \(\text{Re}\,C_1p^{3/2}>0\) for \(\text{Re}\,p=0\). The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem and to find the main term of the asymptotic representation of solutions.

MSC:

35K55	Nonlinear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35C20	Asymptotic expansions of solutions to PDEs
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

initial-boundary value problem; Ott-Sudan-Ostrovskiy type equation; global existence; asymptotic representation