Existence and stability of solutions with periodically moving weak internal layers

We consider the periodic parabolic differential equation $ep^2 Big( fracpartial^2 upartial x^2 -fracpartial upartial t Big)=f(u,x,t,ep)$ under the assumption that $ve$ is a small positive parameter and that the degenerate equation $f(u,x,t,0) =0$ has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution $u_p(x,t,ep)$ which is $T$-periodic in $t$, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as $eprightarrow 0$

Singularly perturbed partly dissipative reaction–diffusion systems in case of exchange of stabilities

We consider the singularly perturbed partly dissipative reaction-diffusion system ε2 (∂u ⁄ ∂t - ∂2u ⁄ ∂x2 = g(u,v,x,t,ε), ∂v ⁄ ∂t = ƒ(u,v,x,t,ε) under the condition that the degenerate equation g(u,v,t,0) = 0 has two solutions u = φi(v,x,t), i = 1,2, that intersect (exchange of stabilities). Our main result concerns existence and asymptotic behavior in ε of the solution of the initial boundary value problem under consideration. The proof is based on the method of asymptotic lower and upper solutions

Singularly perturbed partly dissipative reaction-diffusion systems in case of exchange of stabilities

SIGLEAvailable from TIB Hannover: RR 5549(572)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Singularly perturbed boundary value problems in case of exchange of stabilities

We consider a mixed boundary value problem for a system of two second order nonlinear differential equations where one equation is singularly perturbed. We assume that the associated equation has two intersecting families of equilibria. This property excludes the application of standard results. By means of the method of upper and lower solutions we prove the existence of a solution of the boundary value problem and determine its asymptotic behavior with respect to the small parameter. The results can be used to study differential systems modelling bimolecular reactions with fast reaction rates. (orig.)Available from TIB Hannover: RR 5549(379) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman