We are concerned with the qualitative analysis of positive singular solutions with blow-up boundary for a class of logistic-type equations with slow diffusion and variable potential. We establish the exact blow-up rate of solutions near the boundary in terms of Karamata regular variation theory. This enables us to deduce the uniqueness of the singular solution.
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.
In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: $ u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)+ru-μu^k$, $0=Δv-v+u$ under the non-flux boundary conditions in a smooth bounded convex domain $Ω\subset\mathbb{R}^n$, $χ,r,μ>0$, $k>1$ and $n\ge 2$. It is shown that the system possesses a globally bounded classical solution if $k>\frac{3n-2}{n}$, and $r>\frac{χ^2}{4}$ for $0<χ\le 2$, or $r> χ-1$ for $χ>2$. In addition, under the same condition for $r,χ$, the system admits a global generalized solution when $k\in(2-\frac{1}{n},\frac{3…
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, …
Assuming that $0<χ<\sqrt{\frac{2}n}$, $κ\ge 0$ and $μ>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing \[ \begin{split} u_t &= Δu - χ\nabla\cdot ( \frac{u}{v} \nabla v ) + κu -μu^2\\ v_t &= Δv - u v \end{split} \] in a bounded domain $Ω\subset \mathbb{R}^n$, with homogeneous Neumann boundary conditions and for widely arbitrary positive initial data. In the spatially one-dimensional setting, we prove global existence and, moreover, boundedness of the solution for any $χ>0$, $μ>0$, $κ\ge 0$.
This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t + u\cdot\nabla n =Δn^m - χ\nabla\cdot(n\nabla c) + κn -μn^2$, $x \in Ω,\ t>0$, $c_t + u\cdot\nabla c = Δc - nc$, $x \in Ω,\ t>0$, $u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ, \quad \nabla\cdot u = 0$, $x \in Ω,\ t>0$, where $Ω\subset \mathbb{R}^3$ is a bounded domain and $χ,κ\ge 0$ and $m, μ>0$. In the above system without fluid environment Jin (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with $m=1…
This paper focuses on the following Keller-Segel system with singular sensitivity and logistic source $$ \left\{\begin{array}{ll} u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)+ au-μu^2,\quad x\in Ω, t>0, \disp{ v_t=Δv- v+u},\quad x\in Ω, t>0 \end{array}\right.\eqno(\star) $$ in a smoothly bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, with zero-flux boundary conditions, where $a>0,μ>0$ and $χ>0$ are given constants. If $χ$ is small enough, then, for all reasonable regular initial data, a corresponding initial-boundary value problem for $(\star)$ possesses a global classical solution $(u, v)$ which …
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.
The work is devoted to the construction of the asymptotic behavior of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends to zero. The asymptotics of the solution of such problems contains boundary layer 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 …
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 the differentiability of mild solutions with respect to $\varepsilon$. The operator $G$ can be a singular perturbation of $A$, in the sense that its domain can be strictly contained in the domain of $A$.
In this work, we prove the existence of a family of solutions of the Allen-Cahn equation with nonlinear Neumann boundary condition under some constraints, whose nodal sets concentrate asymptotically to a given volume nondegenerate capillary hypersurface in a compact Riemannian manifold. Our construction is inspired by the works of Pacard and of Pacard and Ritoré.
The paper presents the solution for the existence of analytic solutions for some generalized Lane-Emden (LE) equation. Such solutions exists on the unit interval, which endpoints are singularities of the proposed perturbed LE equation. The solution has many possible applications and one of the examples was provided.
The second-order singularly-perturbed problem concerns the integral functional $\int_Ω\varepsilon_n^{-1}W(u) + \varepsilon_n^3\|\nabla^2u\|^2\,dx$ for a bounded open set $Ω\subseteq \mathbb{R}^N$, a sequence $\varepsilon_n \to 0^+$ of positive reals, and a function $W:\mathbb{R} \to [0,\infty)$ with exactly two distinct zeroes. This functional is of interest since it models the behavior of phase transitions, and its Gamma limit as $n \to \infty$ was studied by Fonseca and Mantegazza. In this paper, we study an instance of the problem for $N=1$. We find a different form for the Gamma limit, and…
