Stability with respect to pseudodifferential perturbations of some nonlinear diffusive equations (Q2706204)

Let \(g^{\varepsilon }(k)\) be an even function of \(k\in \mathbb{Z}\) which satisfies the inequality NEWLINE\[NEWLINEg^{\varepsilon }(k)\leq \rho -\nu \left|k\right|^{2},\quad k\in\mathbb{Z}. NEWLINE\]NEWLINE Assume, moreover, that for all \(k,g^{\varepsilon }(k)\) tends to a limit \( g^{0}(k)\) as \(\varepsilon \rightarrow 0.\) Define a linear operator \( L^{\varepsilon }\) on periodic functions of period \(2\pi \) by NEWLINE\[NEWLINE \left( L^{\varepsilon }u\right)\widehat(k)=g^{\varepsilon }(k)\widehat{u} (k). NEWLINE\]NEWLINE The author proves that the solution of the perturbed equation NEWLINE\[NEWLINE u_{t}^{\varepsilon }=L^{\varepsilon }u^{\varepsilon }-\frac{\left( u_{x}^{\varepsilon }\right) ^{2}}{2} NEWLINE\]NEWLINE converges in a some sense to the solution of the nonperturbed equation as \( \varepsilon =0.\) This result has a number of applications in the analysis of nonlinear phenomena and particularly of combustion phenomena.