$$L^2$$ L 2 -energy decay of convective nonlinear PDEs reaction–diffusion systems via auxiliary ODEs systems

Abstract

The \(L^2\)-energy longtime behaviour of convective binary P.D.Es. reaction–diffusion systems under the action of: self and cross diffusion; nonlinear reaction terms and Robin boundary conditions, is investigated. An auxiliary ODEs system, depending on a positive parameter \(\mu \), is introduced. Via the energy decay of the auxiliary system, for general classes of nonlinear reaction terms, the absence of subcritical instabilities and the \(L^2\)-energy asymptotic decay, are obtained. Estimates of the attraction basin are furnished. Instead of Sobolev type inequalities, only algebraic inequalities are requested. Applications to a celebrated model are furnished.