In this paper we study the asymptotic nonlinear dynamics of scalar semilinear parabolic problems of reaction-diffusion type when the diffusion coefficient becomes large in a subregion in the interior to the domain. We obtain, under suitable assumptions, that the family of attractors behaves continuously and we exhibit the rate of convergence. An accurate description of the localized large diffusion is necessary.

This paper deals with a quasilinear chemotaxis system with an external signal consumption \begin{equation*} \begin{cases} u_t=\nabla\cdot(\varphi(u)\nabla u)-\nabla\cdot(u\nabla v), & (x,t)\in \Omega\times (0,\infty), \\ 0=\Delta v+u-g(x), &(x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a ball $\Omega\subset \mathbb{R}^{n}$, where $\varphi(u)$ is a nonlinear diffusion function and $g(x)$ is an external signal consumption. Under suitable assumptions on the functions $\varphi$ and $g$, it is proved that there exists initial data such that the solution of the above system blows up in finite time.

We provide sufficient conditions for a uniform $L^{2^\star}(\Omega)$ bound to imply a uniform $L^\infty (\Omega)$ bound for positive classical solutions to a class of subcritical elliptic problems in bounded $C^2$ domains in $\mathbb R^N$. We also establish an equivalent result for sequences of boundary value problems.

In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a Conley index for families of nonautonomous evolution operators. Different variants such as a categorial index, a homotopy index and a homology index are obtained. Furthermore, attractor-repeller decompositions and connecting homomorphisms are introduced for the nonautonomous setting.

In previous works the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. In this paper, the treatment of attractor-repeller decomposition is refined. The more general concept of partially ordered Morse-decompositions is used. It is shown that, in the nonautonomous setting, these Morse-decompositions persist under small perturbations. Furthermore, a continuation property for these Morse decompositions is established. Roughly speaking, the index of every Morse-set and every connecting homomorphism continue as the nonautonomous problem, depending continuously on a parameter, changes.

Stimulated by recent problems in the theory of iterated function systems, we provide a variant of the Banach converse theorem for multivalued maps. In particular, we show that attractors of continuous multivalued maps on metric spaces are stable. Moreover, such attractors in locally compact, complete metric spaces may be obtained by means of the Banach theorem in the hyperspace.

In this paper for any $\varepsilon >0$ we construct a new proper $k$-ball-contractive retraction of the closed unit ball of the Banach space $C^m [0,1]$ onto its boundary with $k < 1+ \varepsilon$, so that the Wośko constant $W_\gamma (C^m [0,1])$ is equal to $1$.

In this work, we establish necessary and sufficient conditions for the uniform and orbital stability of a special class of sets on impulsive dynamical systems. The results are achieved by means of Lyapunov functions.

In this paper, we consider the following problem involving fractional Laplacian operator \begin{equation*} (-\Delta)^{s} u=\lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{s}-2}u\quad \text{in } \Omega,\qquad u=0\quad \text{on } \partial\Omega, \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $0<s<1$, $2^*_{s}={2N}/({N-2s})$, and $(-\Delta)^{s}$ is the fractional Laplacian. We will prove that there exists $\lambda_{*}>0$ such that the problem has at least two positive solutions for each $\lambda\in (0,\lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.

We consider the multiplicity and concentration of solutions for the Kirchhoff Type Equation \[ -\varepsilon^2 M\bigg( \varepsilon^{2-N}\int_{\mathbb{R}^N} |\nabla v|^2dx \bigg) \Delta v+V(x)v=f(v), \quad \mathrm{in }\ \mathbb{R}^N. \] Under suitable conditions on functions $M$, $V$ and $f$, we obtain the existence of positive solutions concentrating around the local maximum points of $V$, which gives an affirmative answer to the problem raised in [21]. Moreover, we also obtain multiplicity of solutions which are affected by the topology of critical points set of potential $V$.

This paper studies the existence of global weak solutions for reaction-diffusion systems depending on two main assumptions: the non-negative of solutions and the total mass of components are preserved with time, the non-linearities have critical growth with respect to the gradient. This work is a generalization of the work developed by Alaa and Lefraich [2] without the presence of the gradient in the kinetic reaction terms.

Let $M$ be a connected compact $C^\infty$ manifold of dimension $n\ge4$ without boundary. Let $ \mathcal{M}^k$ be the set of all $C^k$ Riemannian metrics on $M$. Any $g\in\mathcal{M}^k$ determines the Weyl tensor $$ \mathcal W^g\colon M\to \mathbb R^{4n},\qquad \mathcal W^g(\xi):=(W^g_{ijkl}(\xi))_{i,j,k,l=1,\dots,n}.$$ We prove that the set $$\mathcal{A}:=\big\{g\in \mathcal{M}^k : |\mathcal W^g(\xi)|+|D \mathcal W^g(\xi)|+|D^2 \mathcal W^g(\xi)|>0\ \hbox{for any}\ \xi\in M\big\}$$ is an open dense subset of $\mathcal{M}^k$.

In this article, we study the existence and uniqueness of fixed points for some mixed monotone operators and monotone operators with perturbation. These mixed monotone operators and monotone operators are $e$-concave-convex operators and $e$-concave operators respectively. Without using compactness or continuity, we obtain the existence and uniqueness of fixed points by monotone iterative techniques and properties of cones. Our main results extended and improved some existing results. Also, we applied the results to some differential equations.

In this paper we prove the existence of positive ground state solution for a class of linearly coupled systems involving Kirchhoff-Schrödinger equations. We study the subcritical and critical case. Our approach is variational and based on minimization technique over the Nehari manifold. We also obtain a nonexistence result using a Pohozaev identity type.

In this paper, we present an overlapping generation model (OLG for short) of resource extraction with a random production function and an altruism having both paternalistic and non-paternalistic features. All generations are risk-sensitive with a constant coefficient of absolute risk aversion. The preferences are represented by a possibly dynamic inconsistent dynamic recursive utility function with non-cooperating generations. Under general conditions on the aggregator and transition probability, we examine the existence and the uniqueness of a recursive utility function and the existence of a stationary mixed Markov Perfect Nash Equilibria.

We study positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} - \Delta v = \lambda v(1-v), & x \in \Omega_0, \\ \frac{\partial v}{\partial \eta} + \gamma \sqrt{\lambda} ( v-A)^2 v =0 , & x \in \partial \Omega_0, \end{cases} \end{equation*} where $\Omega_0$ is a bounded domain in $\mathbb{R}^n$; $n \ge 1$ with smooth boundary $\partial \Omega_0$, ${\partial }/{\partial \eta}$ is the outward normal derivative, $A \in (0,1)$ is a constant, and $\lambda$, $\gamma$ are positive parameters. Such models arise in the study of population dynamics when the population exhibits a U-shaped density dependent dispersal on the boundary of the habitat. We establish existence, multiplicity, and uniqueness results for certain ranges of the parameters $\lambda$ and $\gamma$. We obtain our existence and mulitplicity results via the method of sub-super solutions.

Adapting methods introduced by Steven Smale, we describe good strategies for a symmetric version of the Iterated Prisoner's Dilemma with $n$ players.

R. Miculescu and A. Mihail in 2008 introduced the concept of a generalized iterated function system (GIFS in short), a particular extension of the classical IFS. The idea is that, instead of families of selfmaps of a metric space $X$, GIFSs consist of maps defined on a finite Cartesian $m$-th power $X^m$ with values in $X$ (in such a case we say that a GIFS is of order $m$). It turned out that a great part of the classical Hutchinson theory has natural counterpart in this GIFSs' framework. On the other hand, there are known only few examples of fractal sets which are generated by GIFSs, but which are not IFSs' attractors. In the paper we study $0$-dimensional compact metrizable spaces from the perspective of GIFSs' theory. Such investigations for classical IFSs have been undertaken in the last several years, for example by T. Banakh, E. D'Aniello, M. Nowak, T.H. Steele and F. Strobin.

We prove that each such space $X$ is homeomorphic to the attractor of some GIFS on the real line. Moreover, we prove that $X$ can be embedded into the real line $\mathbb R$ as the attractor of some GIFS of order $m$ and (in the same time) a nonattractor of any GIFS of order $m-1$, as well as it can be embedded as a nonattractor of any GIFS. Then we show that a relatively simple modifications of $X$ deliver spaces whose each connected component is ''big'' and which are GIFS's attractors not homeomorphic with IFS's attractors. Finally, we use obtained results to show that a generic compact subset of a Hilbert space is not the attractor of any Banach GIFS.