Internal Layer Solutions in Quasilinear Integro-Differential Equations

The Dirichlet boundary value problem for a class of singularly perturbed quasilinear integro-differential equations is considered. The asymptotic expansion for a new class of solutions, which have internal layers, is constructed. Theorems on existence, local uniqueness and asymptotic stability of such internal layer solutions are proved

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

Immediate exchange of stabilities in singular perturbed systems

We study the initial value problem for singularly perturbed systems of ordinary differential equations whose associated systems have two transversally intersecting families of equilibria (transcritical bifurcation) which exchange their stabilities. By means of the method of upper and lower solutions we derive a sufficient condition for the solution of the initial value problem to exhibit an immediate exchange of stabilities. Concerning its asymptotic behavior with respect to #epsilon# we prove that an immediate exchange of stabilities implies a change of the asymptotic behavior from 0(#epsilon#) to 0(#sq root#(#epsilon#)) near the point of exchange of stabilities. (orig.)Available from TIB Hannover: RR 5549(363)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Delayed exchange of stabilities in singularly perturbed systems

We consider a scalar nonautonomous singularly perturbed differential equation whose degenerate equation has two solutions which intersect at some point. These solutions represent families of equilibria of the associated equation where at least one of these families loses its stability at the intersection point. We study the behavior of the solution of an initial value problem of the singularly perturbed equation in dependence on the small parameter. We assume that the solution stays at the beginning near a stable branch of equilibria of the associated system where this branch loses its stability at some critical time t_c. By means of the method of upper and lower solutions we determine the asymptotic delay t* of the solution for leaving the unstable branch. The obtained result holds for the case of transcritical bifurcation as well as for the case of pitchfork bifurcation. We consider some examples where we prove that a well-known result due to N.R. Lebovitz and R.J. Schaar about an immediate exchange of stabilitis cannot be applied to singularly perturbed systems whose right hand side depends on #epsilon#. (orig.)Available from TIB Hannover: RR 5549(270)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

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