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We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order $O(sqrt{epsilon})$ in the $L^infty(L^2)$-norm with diffusion $epsilon to 0$. In this paper, we prove a corresponding result for problems on one-dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a-priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.
Given a symmetric Riemannian manifold (M, g), we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number {epsilon} and the symm
We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations u u_{tt}+u_t=D u+f(u)+ u^alphadot{W} on a bounded spatial domain. An asymptoti
We consider semilinear stochastic evolution equations on Hilbert spaces with multiplicative Wiener noise and linear drift term of the type $A + varepsilon G$, with $A$ and $G$ maximal monotone operators and $varepsilon$ a small parameter, and study t
We consider a family of linear singularly perturbed PDE relying on a complex perturbation parameter $epsilon$. As in a former study of the authors (A. Lastra, S. Malek, Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, J
An averaging method is applied to derive effective approximation to the following singularly perturbed nonlinear stochastic damped wave equation u u_{tt}+u_t=D u+f(u)+ u^alphadot{W} on an open bounded domain $DsubsetR^n$,, $1leq nleq 3$,. Here $ u>0