Stability with Respect to Pseudodifferential Perturbations of Some Nonlinear Diffusive Equations

Abstract

Let $g^\ve(k)$ be an even function of $k\in\Zed$ which satisfies the inequality \forall k \in \Zed, \quad g^\ve(k)\le \rho -\nu \abs{k}^2. Assume, moreover, that $\forall k$, $g^\ve(k)$ tend to a limit g0 (k) as $\ve$ tends to 0. Define a linear operator $\lv$ on periodic functions of period $2\pi$ by \bigl(\lv u\bigr)\,\,\hat{}\; (k) = g^\ve(k)\hat u(k). Joulin has asked whether the solution of \uv_t =\lv \uv -\bigl(\uv_x\bigr)^2 /2 converges to the solution of the analogous problem for $\ve=0$. It is proved here that the answer is positive. Such a positive answer is a means of validating a number of theoretical procedures in the analysis of nonlinear phenomena and particularly of combustion phenomena.