On the eigenfunction expansion method for semilinear dissipative equations in bounded domains and the Kuramoto--Sivashinsky equation in a ball (Q2773402)

The paper consists of 6 sections. In section 1 the method of solution of the abstract Cauchy problem NEWLINE\[NEWLINEu'(t)+ Au(t)=B \bigl(u(t),u(t) \bigr),\;t>0,\;u(0)=0 \tag{1}NEWLINE\]NEWLINE is presented, where \(u(t)\) is a continuous function for \(t\geq 0\) with values in a Hilbert space, and \(B(\cdot,\cdot)\) is a bilinear form. For small initial data the solution in the form of a Fourier series with respect to the eigenvectors of \(A\) is obtained. The basic ideas of this method were developed in the author's earlier papers concerning the Boussinesq equation and the nonlinear heat equation (quoted in the references at the end). Further sections of the reviewed paper are devoted to the Kuramoto-Sivashinsky equation NEWLINE\[NEWLINE\partial_t u+\nu\Delta^2 u+\Delta u=|\nabla u|^2 \tag{2}NEWLINE\]NEWLINE in three spatial dimensions. After introducing basic definitions needed in the sequel (section 2) the first initial-boundary value problem for (2) in the unit ball is examined in section 3. The two theorems claim the following: for \(\nu>1/ \pi^2\) and sufficiently small initial data equation (2) has an unique solution which can be represented as a series (Theorem 1), and this solution has an exponential decay when \(t\to\infty\) (Theorem 2). The proofs of both theorems are presented in sections 5 and 6 after proving in section 4 several auxiliary results.