Initial and Boundary Blow-U Problem for $$$$-Lalacian Parabolic Equation with General Absortion

In this article, we investigate the initial and boundary blow-up problem for the \(p\)-Laplacian parabolic equation \(u_t-\Delta _p u=-b(x,t)f(u)\) over a smooth bounded domain \(\Omega \) of \(\mathbb {R}^N\) with \(N\ge 2\), where \(\Delta _pu=\mathrm{div}(|\nabla u|^{p-2}\nabla u)\) with \(p>1\), and \(f(u)\) is a function of regular variation at infinity. We study the existence and uniqueness of positive solutions, and their asymptotic behaviors near the parabolic boundary.     

Exact solution and invariant for fractional Cattaneo anomalous diffusion of cells in two-dimensional comb framework

This paper presents an investigation on anomalous diffusion of cells in a two-dimensional comb framework with effects of fractional Cattaneo flux. Formulated governing equation is an evolution equation with the coexisting characteristics of parabolic (diffusion) and hyperbolic (wave) for \(\alpha \) in (0, 1). Exact solution is obtained by the special fractional integral transformations, and a novel invariant is established, i.e., \(\left\langle {x^{2}\left( t \right) } \right\rangle \cdot \left\langle P \right\rangle = 0.5\) (the mean square displacement multiplied by the total number of cells along the x-axis = 0.5). Moreover, the characteristics of cells distribution, the total number and the mean square displacement of cells along the x-axis with different involved parameters, especially with the fractional parameter evolution, are shown graphically and analyzed in detail. For the cells distribution versus x, it turns from parabolic and hyperbolic with the decrease in t or the increase in \(\alpha \) or \(\xi \). It is monotonically decreasing for the cells distribution versus \(\alpha \) with different x, t and \(\xi \). For the distribution versus t with different \(\alpha \) and \(\xi \) or versus \(\alpha \) with different t, it is monotonically decreasing for the distribution of total number while monotonically increasing for the distribution of mean square displacement. It is remarkable that the anomalous subdiffusion happens along the x-axis for arbitrary parameters which is different from the classical Cattaneo diffusion.     

Biorthogonal stretching and shearing of an impermeable surface in a uniformly rotating fluid system

The flow induced by an impermeable flat surface executing orthogonal stretching and orthogonal shearing in a rotating fluid system is investigated. Both the stretching and shearing are linear in the coordinates. An exact similarity reduction of the Navier–Stokes equations gives rise to a pair of nonlinearly-coupled ordinary differential equations governed by three parameters. In this study we set one parameter and analyze the problem which leads to flow for an impermeable surface with shearing and stretching due to velocity u along the x-axis of equal strength a while the shearing and stretching due to velocity v along the y-axis of equal strength b. These solutions depend on two parameters—a Coriolis (rotation) parameter \(\sigma = \Omega /a\) and a stretching/shearing ratio \(\lambda =b/a\). A symmetry in solutions is found for \(\lambda = 1\). The exact solution for \(\sigma = 0\) and the asymptotic behavior of solutions for \(|\sigma | \rightarrow \infty\) are determined and compared with numerical results. Oscillatory solutions are found whose strength increases with increasing values of \(|\sigma |\). It is shown that these solutions tend to the well-known Ekman solution as \(|\sigma | \rightarrow \infty\).     

Chaotic characteristics analysis of the sintering process system with unknown dynamic functions based on phase space reconstruction and chaotic invariables

We develop a local discontinuous Galerkin finite element method for the distributed-order time and Riesz space-fractional convection–diffusion and Schrödinger-type equations. The stability of the presented schemes is proved and optimal order of convergence \(\mathcal {O}(h^{N+1}+(\Delta t)^{1+\frac{\theta }{2}}+\theta ^{2})\) for the Riesz space-fractional diffusion and Schrödinger-type equations with distributed order in time, an order of convergence of \(\mathcal {O}(h^{N+\frac{1}{2}}+(\Delta t)^{1+\frac{\theta }{2}}\) \(+\theta ^{2})\) is provided for the Riesz space-fractional convection–diffusion equations with distributed order in time where h, \(\theta \) and \(\Delta t\) are space step size, the distributed-order variables and the step sizes in time, respectively. Finally, the performed numerical examples confirm the optimal convergence order and illustrate the effectiveness of the method.     

Models of Elastic Shells in Contact with a Rigid Foundation: An Asymptotic Approach

We consider a family of linearly elastic shells with thickness \(2\varepsilon\) (where \(\varepsilon\) is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface \(S\), and may enter in contact with a rigid foundation along the bottom face.We are interested in studying the limit behavior of both the three-dimensional problems, given in curvilinear coordinates, and their solutions (displacements \(\boldsymbol{u}^{\varepsilon}\) of covariant components \(u_{i}^{\varepsilon}\)) when \(\varepsilon\) tends to zero. To do that, we use asymptotic analysis methods. On one hand, we find that if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), a suitable approximation of the variational formulation of the contact problem is a two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem for an elastic membrane. On the other hand, if the applied body force density is \(O(\varepsilon^{2})\) and surface tractions density is \(O(\varepsilon^{3})\), the corresponding approximation is a different two-dimensional inequality which can be identified as the variational formulation of the obstacle problem for an elastic flexural shell. We finally discuss the existence and uniqueness of solution for the limit two-dimensional variational problems found.     

Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations

We consider bounded solutions of the semilinear heat equation \(u_t=u_{xx}+f(u)\) on \(R\), where \(f\) is of the unbalanced bistable type. We examine the \(\omega \)-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at \(x=\pm \infty \), the \(\omega \)-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of \(f\).     

One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity

We consider positive classical solutions of

$$\begin{aligned} v_t=(v^{m-1}v_x)_x, \qquad x\in {\mathbb {R}}, \ t>0, \qquad (\star ) \end{aligned}$$

in the super-fast diffusion range \(m<-1\). Our main interest is in smooth positive initial data \(v_0=v(\cdot ,0)\) which decay as \(x\rightarrow +\infty \), but which are possibly unbounded as \(x\rightarrow -\infty \), having in mind monotonically decreasing data as prototypes. It is firstly proved that if \(v_0\) decays sufficiently fast only in one direction by satisfying

$$\begin{aligned} v_0(x) \le cx^{-\beta } \qquad \text{ for } \text{ all } ~x>0 \quad \hbox { with some }\quad \beta >\frac{2}{1-m} \end{aligned}$$

and some \(c>0\), then the so-called proper solution of (\(\star \)) vanishes identically in \({\mathbb {R}}\times (0,\infty )\), and accordingly no positive classical solution exists in any time interval in this case. Complemented by some sufficient criteria for solutions to remain positive either locally or globally in time, this condition for instantaneous extinction is shown to be optimal at least with respect to algebraic decay of the initial data. This partially extends some known nonexistence results for (\(\star \)) (Daskalopoulos and Del Pino in Arch Rat Mech Anal 137(4):363–380, 1997) in that it does not require any knowledge on the behavior of \(v_0(x)\) for \(x<0\). Next focusing on the phenomenon of extinction in finite time, we show that in this respect a mass influx from \(x=-\infty \) can interact with mass loss at \(x=+\infty \) in a nontrivial manner. Namely, we shall detect examples of monotone initial data, with critical decay as \(x\rightarrow +\infty \) and exponential growth as \(x\rightarrow -\infty \), that lead to solutions of (\(\star \)) which become extinct at a finite positive time, but which have empty extinction sets. This is in sharp contrast to known extinction mechanisms which are such that the corresponding extinction sets coincide with all of \({\mathbb {R}}\).

     

The Mixed Convection Boundary Layer on a Vertical Melting Front in a Non-Darcian Porous Medium

The effect of surface melting on the dual solutions that can arise in the problem of the mixed convection boundary-layer flow past a vertical surface embedded in a non-Darcian porous medium is considered. The problem is described by M, melting parameter, \(\lambda \), mixed convection parameter, and \(\gamma \), the flow inertia coefficient, numerical results being obtained in terms of these three parameters. It is seen that the melting phenomenon reduces the heat transfer rate and enhances the boundary-layer separation at the solid–liquid interface. Asymptotic solutions for the forced convection, \(\lambda =0\), and free convection, large \(\lambda \), limits are derived.     

Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary

We investigate the influence of a shifting environment on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is homogeneous and favourable, this model was first studied in Du and Lin (SIAM J Math Anal 42:377–405, 2010), where a spreading–vanishing dichotomy was established for the long-time dynamics of the species, and when spreading happens, it was shown that the species invades the new territory at some uniquely determined asymptotic speed \(c_0>0\). Here we consider the situation that part of such an environment becomes unfavourable, and the unfavourable range of the environment moves into the favourable part with speed \(c>0\). We prove that when \(c\ge c_0\), the species always dies out in the long-run, but when \(0<c<c_0\), the long-time behavior of the species is determined by a trichotomy described by (a) vanishing, (b) borderline spreading, or (c) spreading. If the initial population is written in the form \(u_0(x)=\sigma \phi (x)\) with \(\phi \) fixed and \(\sigma >0\) a parameter, then there exists \(\sigma _0>0\) such that vanishing happens when \(\sigma \in (0,\sigma _0)\), borderline spreading happens when \(\sigma =\sigma _0\), and spreading happens when \(\sigma >\sigma _0\).     

Spiky Steady States of a Chemotaxis System with Singular Sensitivity

This paper is concerned with the steady state problem of a chemotaxis model with singular sensitivity function in a one dimensional spatial domain. Using the chemotactic coefficient \(\chi \) as the bifurcation parameter, we perform local and global bifurcation analysis for the model. It is shown that positive monotone steady states exist as long as \(\chi \) is larger than the first bifurcation value \(\bar{\chi }_1.\) We further obtain asymptotic profiles of these steady states, as \(\chi \) becomes large. In particular, our results show that the cell density function forms a spike, which models the important physical phenomenon of cell aggregation.     

A practical synchronization approach for fractional-order chaotic systems

A practical synchronization approach is proposed for a class of fractional-order chaotic systems to realize perfect \(\delta \)-synchronization, and the nonlinear functions in the fractional-order chaotic systems are all polynomials. The \(\delta \)-synchronization scheme in this paper means that the origin in synchronization error system is stable. The reliability of \(\delta \)-synchronization has been confirmed on a class of fractional-order chaotic systems with detailed theoretical proof and discussion. Furthermore, the \(\delta \)-synchronization scheme for the fractional-order Lorenz chaotic system and the fractional-order Chua circuit is presented to demonstrate the effectiveness of the proposed method.     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

Oriented Shadowing Property and $$\Omega $$-Stability for Vector Fields

We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.     

A Note on the Convergence of Singularly Perturbed Second Order Potential-Type Equations

In this paper we study the limit as \(\varepsilon \rightarrow 0\) of the singularly perturbed second order equation \(\varepsilon ^2 \ddot{u}_\varepsilon + \nabla _{\!x} V(t,u_\varepsilon (t))=0\), where V(t, x) is a potential. We assume that \(u_0(t)\) is one of its equilibrium points such that \(\nabla _{\!x}V(t,u_0(t))=0\) and \(\nabla _{\!x}^2V(t,u_0(t))>0\). We find that, under suitable initial data, the solutions \(u_\varepsilon \) converge uniformly to \(u_0\), by imposing mild hypotheses on V. A counterexample shows that they cannot be weakened.     

On Exponential Decay and the Markus–Yamabe Conjecture in Infinite Dimensions with Applications to the Cima System

In this work we extend to infinite dimensions a previous result of Lyascenko given in finite dimensions for a class of Lipschitz functions. In particular, this extension substitutes the class of Lipschitz functions by a class of Hölder functions. Then we consider applications of these results to analyze its relation with the Markus–Yamabe Conjecture in infinite dimensions. We discuss in detail a celebrated example of Cima et al. ( Adv Math 131(2): 453–457, 1997) of a nonlinear system in dimension three whose Jacobian has a unique eigenvalue \(-1\) of multiplicity 3, and yet an explicit unbounded solution as \(t \rightarrow \infty \) exists. We also present explicit solutions of the same equation that tends to 0 as \(t \rightarrow \infty \). Then we look at this conjecture by considering not the properties of the whole system, but instead the properties of some solutions. Finally we present an application in an infinite dimensional Hilbert space, where we use different techniques to study the local and global asymptotic stabilities.     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

Further Dense Properties of the Space of Circle Diffeomorphisms with a Liouville Rotation Number

In continuation of Matsumoto’s paper (Nonlinearity 25:1495–1511, 2012) we show that various subspaces are \(C^{\infty }\)-dense in the space of orientation-preserving \(C^{\infty }\)-diffeomorphisms of the circle with rotation number \(\alpha \), where \(\alpha \in {\mathbb {S}}^1\) is any prescribed Liouville number. In particular, for every odometer \({\mathcal {O}}\) of product type we prove the denseness of the subspace of diffeomorphisms which are orbit-equivalent to \({\mathcal {O}}\).     

Infinitely Many Stability Switches in a Problem with Sublinear Oscillatory Boundary Conditions

We consider the elliptic equation \(-\Delta u +u =0\) with nonlinear boundary condition \(\frac{\partial u}{\partial n}= \lambda u + g(\lambda ,x,u), \) where \(\frac{g(\lambda ,x,s)}{s} \rightarrow 0, \hbox { as }|s|\rightarrow \infty \) and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.