Let $a₁,...,a_k$ be arbitrary nonzero real numbers. An $(a₁,...,a_k)$-decomposition of a function f:ℝ → ℝ is a sum $f₁+\cdots +f_k=f$ where $f_i:ℝ\to ℝ$ is an $a_i$-periodic function. Such a decomposition is not unique because there are several solutions of the equation $h₁+\cdots +h_k=0$ with $h_i:ℝ\to ℝa_i$-periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a₁,...,a_k)$-decomposition is essentially unique. We characterize those periods for which essential...

Let $u\in L^2(I;H^1\left(\Omega \right))$ with ${\partial }_{t}u\in L^2(I;H^1{\left(\Omega \right)}^{*})$ be given. Then we show by means of a counter-example that the positive part $u^{+}$ of $u$ has less regularity, in particular it holds ${\partial }_t{u}^{+}\notin L^1(I;H^1{\left(\Omega \right)}^{*})$ in general. Nevertheless, $u^{+}$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_t-M\left({\int }_{\Omega }\phi u\mathrm{d}x\right)\mathrm{div}\left(A(x,t,u)\nabla u\right)=g(x,t,u)$ in $\Omega \times (0,T)$, where $\Omega$ is a bounded domain of ${ℝ}^n$ $(n\ge 1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n\times n$ matrix of variable coefficients depending on $u$ and $M:ℝ\to ℝ$, $\phi :\Omega \to ℝ$, $g:\Omega \times (0,T)\times ℝ\to ℝ$ are given functions. We consider two different assumptions on $g$. The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if $A(x,t,u)=a(x,t)$ depends only on...

Consider the following higher order difference equation $$x(n+1)=f(n,x(n\left)\right)+g(n,x(n-k\left)\right),n=0,1,\cdots$$ where $f(n,x)$ and $g(n,x):\{0,1,\cdots \}\times [0,\infty )\to [0,\infty )$ are continuous functions in $x$ and periodic functions in $n$ with period $p$, and $k$ is a nonnegative integer. We show the existence of a periodic solution $\left\{\tilde{x}\left(n\right)\right\}$ under certain conditions, and then establish a sufficient condition for $\left\{\tilde{x}\left(n\right)\right\}$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also...

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then a sequence $x\in A^{\mathbb{N}}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately...

We consider $2m$th-order $(m\ge 2)$ semilinear parabolic equations $u_t=-{\left(-\Delta \right)}^{m}u\pm {\left|u\right|}^{p-1}u$ in ${ℛ}^N\times {ℝ}_{+}$ $(p>1)$, with Dirac’s mass $\delta \left(x\right)$ as the initial function. We show that for $p<p_0=1+2m/N$, the Cauchy problem admits a solution $u(x,t)$ which is bounded and smooth for small $t>0$, while for $p\ge p_0$ such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

In this work we study the problem $u_t{-div\left(\right|x|}^{-2\gamma }\nabla u)=\lambda \frac{u^{\alpha }}{{\left|x\right|}^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0\left(x\right)$ in $\Omega$, $\Omega \subset {ℝ}^N$ $(N\ge 2)$ is a bounded regular domain such that $0\in \Omega$, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <\left(N-2\right)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1\left(\Omega \right)$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

In this paper, we propose an extension of a periodic $GARCH$ ($PGARCH$) model to a Markov-switching periodic $GARCH$ ($MS$-$PGA$ $RCH$), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically and of weakly periodically stationary solutions. We establish necessary and sufficient conditions ensuring the existence of higher order moments. We further provide closed-form expressions for calculating the even-order moments as well...

In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $$x_{n+1}=\frac{a{x}_{n-3}}{b+c{x}_{n-1}{x}_{n-3}},n=0,1,\cdots$$ where $a,b,c$ are positive real numbers and the initial conditions $x_{-3}$, $x_{-2}$, $x_{-1}$, $x_0$ are real numbers.

We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-\beta }$, with $\beta \in [0,\infty )$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $\beta$. If $\beta \in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta =1$, it is given by a parabolic equation involving an operator...

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_j$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n,\nu }$ be the space of piecewise linear continuous functions on the torus with knots $t_j:0\le j\le N-1$. Finally, let $P_{n,\nu }$ be the orthogonal projection operator from L²([0,1)) onto $V_{n,\nu }$. The main result is $li{m}_{n\to \infty ,\nu =1}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=su{p}_{n\in \mathbb{N},0\le \nu \le n}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=2+(33-18\surd 3)/13$. This shows in particular that the Lebesgue constant of the classical...

In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S)${}_{\epsilon }$ with a nonnegative constant $\epsilon$, and for any $\epsilon \ge 0$, (S)${}_{\epsilon }$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $\Omega$, and the parabolic equation on the boundary $\Gamma :=\partial \Omega$, having a sufficient smoothness. The objective of this study is to establish a mathematical method,...

We consider the equation $\mathrm{d}y\left(t\right)/\mathrm{d}t=(A+B(t\left)\right)y\left(t\right)$ $(t\ge 0)$, where $A$ is the generator of an analytic semigroup ${\left({\mathrm{e}}^{At}\right)}_{t\ge 0}$ on a Banach space $𝒳$, $B\left(t\right)$ is a variable bounded operator in $𝒳$. It is assumed that the commutator $K\left(t\right)=AB\left(t\right)-B\left(t\right)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\parallel S{\mathrm{e}}^{At}\parallel$ is integrable and the operator $K\left(t\right)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ${\nabla }^{\omega }$ on the corresponding tractor bundle $V$, where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that ${\nabla }^{\omega }$ yields the prolongation of this operator in the homogeneous case $M=G/P$. Our first...

In this work we consider a diffusion problem in a periodic composite having three phases: matrix, fibers and interphase. The heat conductivities of the medium vary periodically with a period of size ${\epsilon }^{\beta }$ ($\epsilon >0$ and $\beta >0$) in the transverse directions of the fibers. In addition, we assume that the conductivity of the interphase material and the anisotropy contrast of the material in the fibers are of the same order ${\epsilon }^2$ (the so-called double-porosity type scaling) while the matrix material has a conductivity...