This paper is concerned with the following singularly perturbed non-local semi-linear problem \begin{equation} \label{h} \tag{$\ast$} \begin{cases} \varepsilon^2 Δu=\frac{m}{\int_Ωe^{u}{\mathrm{d}x}}u e^u\quad &\mathrm{in}~Ω,\\ u= u_0~&\mathrm{on}~\partialΩ, \end{cases} \end{equation} which corresponds to the stationary problem of a chemotaxis system describing the aerobic bacterial movement, where $Ω$ is a smooth bounded domain in $\mathbb{R}^N (N\geq 1)$, $\varepsilon, m$ and $u_0$ are positive constants. We show that the problem \eqref{h} admits a unique classical solution which is of boundary-layer profile as $\varepsilon \to 0$, where the boundary-layer thickness is of order $\varepsilon$. When $Ω=B_R(0)$ is a ball with radius $R>0$, we find a refined asymptotic boundary layer profile up to the first-order expansion of $\varepsilon$ by which we find that the slope of the layer profile in the immediate vicinity of the boundary decreases with respect to (w.r.t.) the curvature while the boundary-layer thickness increases {w.r.t.} the curvature.
A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the solutions to the perturbed problems is studied as $\varepsilon \to 0^+$, recovering classical results in the literature.
A theorem is proved on the uniform estimation of the residual term of the asymptotic expansion with respect to a small parameter of the solution of the initial problem for a singularly perturbed differential operator weakly nonlinear transport equation in the critical case.
An asymptotic expansion with respect to a small parameter of a singularly perturbed system of hyperbolic equations, describing vibrations of two rigidly connected strings is constructed. Under certain conditions imposed on these problems, the principal term of the asymptotic expansion of the solution is described by the Korteweg-de Vries equation.
We study fully nonlinear singularly perturbed parabolic equations and their limits. We show that solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. For the limiting free boundary problem, we analyse the behaviour of solutions near the free boundary. We show, in particular, that, at each time level, the free boundary is a porous set and, consequently, is of Lebesgue measure zero. For rotationally invariant operators, we also derive the limiting free boundary condition.
We consider the following singularly perturbed elliptic problem \[ - {\varepsilon ^2}Δu + u = f(u){\text{ in }}Ω,{\text{ }}u > 0{\text{ in }}Ω,{\text{ }}u = 0{\text{ on }}\partial Ω, \] where $Ω$ is a domain in ${\mathbb{R}^N}(N \ge 3)$, not necessarily bounded, with boundary $\partial Ω\in {C^2}$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a family of multi-peak solutions to the equation given above which concentrate around any prescribed finite sets of local maxima of the distance function from the boundary $\partial Ω$.
We consider the Laplace equation in a domain of $\mathbb{R}^n$, $n\ge 3$, with a small inclusion of size $ε$. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For $ε$ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.
We perform a convergence analysis of a discrete-in-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. We allow for different scalings between the viscosity parameter $\varepsilon$ and the time scale $τ$. When the ratio $\frac{\varepsilon}τ$ diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as formal limit, when $\varepsilon\to 0$, of the gradient flow. We also characterize the limit evolution corresponding to an asymptotically finite ratio betwe…
This work is devoted to study of a class of elliptic singular perturbed systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behaviour with convergence to a limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero which implies simultaneously all components can not coexist. We present a novel method, which provides an explicit solution of limiting problem for special choice of parameters. Moreover, …
In this paper, we aim to investigate the following class of singularly perturbed elliptic problem $$ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\triangle {u}+|x|^ηu =|x|^ηf(u)& \mbox{in}\,\, A, u=0 & \mbox{on}\,\, \partial A, \end{array} \right. $$ where $\varepsilon>0$, $η\in\mathbb{R}$, $A=\{x\in\R^{2N}:\,\,0<a<|x|<b\}$, $N\ge2$ and $f$ is a nonlinearity of $C^1$ class with supercritical growth. By a reduction argument, we show that there exists a nodal solution $u_\e$ with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimens…
In this paper we analyse a boundary value problem for the Laplace equation with a nonlinear non-autonomous transmission conditions on the boundary of a small inclusion of size $ε$. We show that the problem has solutions for $ε$ small enough and we investigate the dependence of a specific family of solutions upon $ε$. By adopting a functional analytic approach we prove that the map which takes $ε$ to (suitable restrictions of) the corresponding solution can be represented in terms of real analytic functions.
Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed $q-$difference-differential equations in the complex domain are constructed. They stand for a $q-$analog of the singularly perturbed partial differential equations considered in our recent work [A. Lastra, S. Malek, Boundary layer expansions for initial value problems with two complex time variables, submitted 2019]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to …
