On the eigenfunction expansion method for semilinear dissipative equations in bounded domains and the Kuramoto–Sivashinsky equation in a ball
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 $C^0([0,\infty ),H_0^s(B))$, $s<2,$ and the uniqueness is proved for $-1+\varepsilon \leq s<2$, where $\varepsilon >0$ is small$.$ The second-order long-time asymptotics is calculated.