On the eigenfunction expansion method for semilinear dissipative equations in bounded domains and the Kuramoto–Sivashinsky equation in a ball

V. V. Varlamov¹

¹ Department of Mathematics The University of Texas at Austin Austin, TX 78712-1082, U.S.A.

Studia Mathematica, Tome 148 (2001) no. 3, pp. 221-249

Résumé

Presented herein is a method of constructing solutions of semilinear dissipative evolution equations in bounded domains. For small initial data this approach permits one to represent the solution in the form of an eigenfunction expansion series and to calculate the higher-order long-time asymptotics. It is applied to the spatially 3D Kuramoto–Sivashinsky equation in the unit ball $B$ in the linearly stable case. A global-in-time mild solution is constructed in the space $C^0([0,\infty ),H_0^s(B))$, $s2,$ and the uniqueness is proved for $-1+\varepsilon \leq s2$, where $\varepsilon >0$ is small$.$ The second-order long-time asymptotics is calculated.

Détail

DOI : 10.4064/sm148-3-3

Keywords: presented herein method constructing solutions semilinear dissipative evolution equations bounded domains small initial approach permits represent solution form eigenfunction expansion series calculate higher order long time asymptotics applied spatially kuramoto sivashinsky equation unit ball linearly stable global in time mild solution constructed space infty uniqueness proved varepsilon leq where varepsilon small second order long time asymptotics calculated