Existence of Generalized Traveling Waves in Time Recurrent and Space Periodic Monostable Equations

This paper is concerned with the extension of the concepts and theories of traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones.  It first introduces the concept of generalized traveling wave solutions of time recurrent and space periodic monostable equations, which extends the concept of periodic traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones. It then proves that in the direction of any unit vector \(\xi\), there is \(c^*(\xi)\) such that for any \(c>c^*(\xi)\), a generalized traveling wave solution in the direction of \(\xi\) with averaged propagation speed \(c\) exists. It also proves that if the time recurrent and space periodic monostable equation is indeed time periodic, then \(c^*(\xi)\) is the minimal wave speed in the direction of \(\xi\) and the generalized traveling wave solution in the direction of \(\xi\) with averaged speed \(c>c^*(\xi)\) is a periodic traveling wave solution with speed \(c\), which recovers the existing results on the existence of periodic traveling wave solutions in the direction of \(\xi\) with speed greater than the minimal speed in that direction.     

On the number of \(n\)-dimensional invariant spheres in polynomial vector fields of \(\mathbb{C}^{n+1}\)

We study the polynomial vector fields \(\mathcal{X}= \displaystyle \sum_{i=1}^{n+1} P_i(x_1,\ldots,x_{n+1}) \frac{\partial}{\partial x_i}\) in \(\mathbb{C}^{n+1}\) with \(n\geq 1\) . Let \(m_i\) be the degree of the polynomial \(P_i\). We call \((m_1,\ldots,m_{n+1})\) the degree of \(\mathcal{X}\). For these polynomial vector fields \(\mathcal{X}\) and in function of their degree we provide upper bounds, first for the maximal number of invariant \(n\)-dimensional spheres, and second for the maximal number of \(n\)-dimensional concentric invariant spheres.     

Bifurcation of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems of degree 5 with a cusp

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian system, which is a Li é nard system of the form \(\dot{x}=y,\)  \(\;\dot{y}=Q_1(x)+\varepsilon yQ_2(x)\) with \(Q_1\) and \(Q_2\) polynomials of degree 4 and 3, respectively. It is shown that this system can undergo degenerated Hopf bifurcation and Poincar é bifurcation, which emerge at most three limit cycles for \(\varepsilon\) sufficiently small.     

Asymptotic behavior of the Cahn-Hilliard-Oono equation

Our aim in this article is to study the asymptotic behavior, in terms of finite-dimensional attractors, of the Cahn-Hilliard-Oono equation. This equation differs from the usual Cahn-Hilliard equation by the presence of a term of the form \( \epsilon u,\ \epsilon >0\), which takes into account long-ranged interactions. In particular, we prove the existence of a robust family of exponential attractors as \(\epsilon\) goes to \(0\).     

Duality theory of regularized resolvent operator family

Let \( k \in C(R^+)\), A be a closed linear densely defined operator in the Banach space \(X\) and \( \{R(t)\}_{t\geq 0} \) be an exponentially bounded \(k\)-regularized resolvent operator families generated by A. In this paper, we mainly study pseudo k-resolvent and duality theory of k-regularized resolvent operator families. The conditions that pseudo k-resolvent become k-resolvent of the closed linear densely defined operator A are given. The some relations between the duality of the regularized resolvent operator families and the generator A are gotten. In addition, the corresponding results of duality of \(k\)-regularized resolvent operator families in Favard space are educed.     

The \(C^1\) solution of the high dimensional Feigenbaum-like functional equation

By constructing a structure operator quite different from before, and using the Schauder's fixed point theory, the existence and uniqueness of the \(C^1\) solutions of the high dimensional Feigenbaum-like functional equations are discussed.     

Analytic Conjugation, Global Attractor, and the Jacobian

It is proved that the dilation \(\lambda f\) of an analytic map \(f\) on \({\bf C}^n$\) with \(f(0)=0,f'(0)=I, |\lambda|>1\) has an analytic conjugation to its linear part \(\lambda x\) if and only if \(f\) is an analytic automorphism on \({\bf C}^n\) and \(x=0\) is a global attractor for the inverse \((\lambda f)^{-1}\). This result is used to show that the dilation of the Jacobian polynomial of [12] is analyticly conjugate to its linear part.     

On the existence of bubble-type solutions of nonlinear singular problems

Considered in this paper is a class of singular boundary value problem, arising in hydrodynamics and nonlinear field theory, when centrally bubble-type solutions are sought: \((p(t)u0)0 = c(t)p(t)f(u); u0(0) = 0; u(+1) = L > 0\) in the half-line \([0;+1)\), where \(p(0) = 0\). We are interested in strictly increasing solutions of this problem in \([0;1)\) having just one zero in \((0;+1) \)and finite limit at zero, which has great importance in applications or pure and applied mathematics. Su±cient conditions of the existence of such solutions are obtained by applying the critical point theory and by using shooting argument [9,10] to better analysis the properties of certain solutions associated with the singular di®erential equation. To the authors' knowledge, for the first time, the above problem is dealt with when f satis¯es non-Lipschitz condition. Recent results in the literature are generalized and signi¯cantly improved.     

Global Stability for a Dynamic model of Hepatitis B with Antivirus Treatment

An epidemic model on the basis of therapy of chronic Hepatitis B with antivirus treatment was introduced in this paper. By applying a comparison theorem and analyzing the corresponding characteristic equations, we obtain sufficient conditions on the parameters for the global stability of the disease-free state. It's proved that if the basic reproduction number \(R_0 < 1\) , the disease-free equilibrium is globally asymptotically stable. If \(R_0 > 1\), the disease-free equilibrium is unstable and the disease is uniformly permanent. Moreover, if \(R_0 > 1\), sufficient conditions are obtained for the global stability of the endemic equilibrium.     

Applications of the group analysis of differential equations to some systems of noncommuting <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth vector fields

Given a canonical basis of C ¹-smooth vector fields \(\{ \tilde X_i \} \) satisfying certain restrictions on commutators, we prove an existence theorem for their local nilpotent homogeneous approximation at the origin using the methods of the group analysis of differential equations. We study the properties of the quasimetrics induced by some systems of vector fields related to \(\{ \tilde X_i \} \).     

Group closures of partial transformations

Let \(X\) be an infinite set, \(f\) a partial one-to-one transformation of \(X\), and \(H\) a normal subgroup of G _X , the group of all permutations of \(X\). We investigate when \(H\) is equal to \(G_{<f:H>}\). That is, we are interested when \(H\) is the full group of normalizers of the semigroup of transformations on \(X\) generated by conjugates of \(f\) by elements of \(H\).     

Cauchy problem for the Zakharov System Arising from Ion-Acoustic Modes with low regularity data

We prove local well-posedness results for the Zakharov System Arising from Ion-Acoustic Modes in two spacial dimension with large initial data in low regularity Sobolev space \(&nbsp;&nbsp; (\dot{H}^1 \bigcup H^{\frac{1}{2}})\times L^2 \times H^{-1}&nbsp; \).&nbsp;&nbsp;Using &rdquo;derivative sharing&rdquo;, the local well-posedness results in \( (\dot{H}^1 \bigcup H^{\frac{1}{2}-\delta})\times H^{\delta} \times H^{-1+\delta}\)&nbsp;&nbsp;are also obtained, for any &nbsp;0&nbsp;\(\leq \delta \leq&nbsp;1/2 \).     

Existence of global solutions of multicomponent reactive transport problems with mass action kinetics in porous media

We prove the existence and uniqueness of time-global solutions for multi-species&nbsp;multi-reaction advection-diffusion-dispersion problems with mass action kinetics&nbsp;in the space \(W_p^{2,1}([0,T]\!\times\!\Omega)\). The reaction terms of mass&nbsp;action kinetics may contain polynomial expressions of arbitrarily high order. The&nbsp;difficulty to obtain an a~priori estimate for the semilinar system of PDEs is&nbsp;tackled with a special Lyapunov function.     

Anti-periodic mild solutions to semilinear fractional differential equations

In this paper, by using the \(\alpha \)-resolvent family theory, Banach contraction mapping principle and Schauder’s fixed point theorem, we investigate the existence of anti-periodic mild solutions to the semilinear fractional differential equations \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t)),\ t\in R,1 \le \alpha \le 2 \) and \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t),u'(t)),\ t\in R,1 < \alpha < 2\), where \(A : D(A)\subset X \rightarrow X\) is the infinitesimal generator of an \(\alpha \)-resolvent family defined on a Banach space \(X\) and \(f\) is a suitable function. Furthermore, an example is given to illustrate our results.     

Existence-Uniqueness Problems For Infinite Dimensional Stochastic Differential Equations With Delays

The main aim of this paper is to develop the basic theory of a class of infinite dimensional stochastic differential equations with delays (IDSDEs) under local Lipschitz conditions. Firstly, we establish a global existence-uniqueness theorem for the IDSDEs under the global Lipschitz condition in \(C\) without the linear growth condition. Secondly, the non-continuable solution for IDSDEs is given under the local Lipschitz condition in \(C\). Then, the classical Itô's formula is improved and a global existence theorem for IDSDEs is obtained. Our new theorems give better results while conditions imposed are much weaker than some existing results. For example, we need only the local Lipschitz condition in \(C\) but neither the linear growth condition nor the continuous condition on the time \(t\). Finally, two examples are provided to show the effectiveness of the theoretical results.     

Method of characteristics and first integrals for systems of quasi-linear partial differential equations of first order

Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.     

Cyclicity of several quadratic reversible systems with center of genus one

By using the Chebyshev criterion to study the number of zeros of Abelian integrals, developed by&nbsp;M. Grau, F. Ma\(\~n\)osas and J. Villadelprat in [2], we prove&nbsp;that the cyclicity of period annulus of the quadratic reversible systems with center of genus one, classified as (r8), (r13) and&nbsp;(r16) by S. Gautier, L. Gavrilov and I. D. Iliev in [1],&nbsp;under quadratic perturbations is two. These results partially give a&nbsp;positive answer to the conjecture 1 in [1].     

Partition regularity of polynomial systems near zero

Semigroup Forum - Recently, S.&nbsp;Kanti Patra and Md.&nbsp;Moid Shaikh proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense...     

Maximal Operators in the Spherical Mean Setting

In this paper, we prove an existence result for \(\mathcal {L}^{\infty }\)-solutions for a class of semilinear delay evolution inclusions with measures and subjected to nonlocal initial conditions of the form

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \mathrm{d}u(t)= \{Au(t)+f(t)\}\mathrm{d}t+\mathrm{d}h(t),&{}\quad t\in \mathbb {R}_+,\\ \displaystyle f(t)\in F(t,u_t),&{}\quad t\in \mathbb {R}_+,\\ \displaystyle u(t)=g(u)(t),&{}\quad t\in [\,-\tau ,0\,]. \end{array} \right. \end{aligned}$$

Here \(\tau \ge 0\), X is a Banach space, \(A:D(A)\subseteq X \rightarrow X \) is the infinitesimal generator of a \(C_0\)-semigroup, \(F:\mathbb {R}_+\times \mathcal {R}([\,-\tau ,0\,];X)\rightsquigarrow X\) is a u.s.c. multifunction with nonempty, convex and weakly compact values, \(h\in BV_{\mathrm{loc}}(\mathbb {R}_+;X)\) and the function \(g:\mathcal {R}_{b}(\mathbb {R}_+;X)\rightarrow \mathcal {R}([\,-\tau ,0\,];X)\) is nonexpansive.

     

Coupled systems of non-smooth differential equations

We study the geometric qualitative behavior of a class of discontinuous vector fields in four dimensions. Explicit existence conditions of one-parameter families of periodic orbits for models involving two coupled relay systems are given. We derive existence conditions of one-parameter families of periodic solutions of systems of two second order non-smooth differential equations. We also study the persistence of such periodic orbits in the case of analytic perturbations of our relay systems. These results can be seen as analogous to the Lyapunov Centre Theorem.