Decay estimates for the solutions of some nonlinear evolution equations
The paper is devoted to decay estimates for one-dimensional Cauchy problems for certain nonlinear evolution equations and systems, as e.g. \(u_ t - \Delta u + (\varphi (u))_ x = 0\) or a KdV-Burgers-type equation. The existence proof of global solutions is omitted; the author refers to a paper by \textit{Y. Zhou} and \textit{B. Guo} [Acta Math. Sin. 27, 154-176 (1984; Zbl 0543.35093)], with the remark, that a proof runs along the same lines. Now the common feature of the equations considered is the derivation of energy-inequalities, which read after Fourier transformation \[ \bigl \| \widehat v(t, \xi) \bigr \|^ 2_ 2 + \int^ t_ 0 \bigl \| \widehat v(s,\xi) | \xi |^ k \bigr \|^ 2_ 2 ds \leq C \quad \text{for some } k, \] where \(v\) denotes the solution \(u\) or its derivative \({\partial u / \partial x}\). Taking this as a starting point, a method of \textit{M. E. Schonbek}, which was successfully applied to the Navier-Stokes equations [Arch. Ration. Mech. Anal. 88, 209-222 (1985; Zbl 0602.76031)] can be used -- a splitting of the integration in phase- space, combined with a representation of the solution in the Fourier- picture for small \(| \xi |\) results in a polynomial decay rate for \(\| v \|_ 2\) with respect to time. The exponent depends on the integrability of the initial value and the vanishing order of the nonlinearity. The author follows this program and proves decay results under certain assumptions on \(\varphi\).
scientific article; zbMATH DE number 729097