Existence and uniqueness of solutions for a class of initial value problems of fractional differential systems on half lines

In this article, we establish the existence and uniqueness results for solutions of a class of initial value problems of nonlinear fractional differential systems on half lines involving Riemann–Liouville fractional derivatives. Our analysis relies on the well-known fixed point theorem of Schauder. The novelty of this paper is that the problems discussed are defined on half lines, and the nonlinearities f and g   are allowed to be singular functions. Furthermore, we allow p∈(0,β)$p\in (0,\beta )$ and q∈(0,α)$q\in (0,\alpha )$.     

Asymptotic patterns of a structured population diffusing in a two-dimensional strip

In this paper, we derive a population model for the growth of a single species on a two-dimensional strip with Neumann and Robin boundary conditions. We show that the dynamics of the mature population is governed by a reaction–diffusion equation with delayed global interaction. Using the theory of asymptotic speed of spread and monotone traveling waves for monotone semiflows, we obtain the spreading speed c^∗$c^{\ast }$, the non-existence of traveling waves with wave speed 0<c<c^∗$0<c<c^{\ast }$, and the existence of monotone traveling waves connecting the two equilibria for c≥c^∗$c\ge c^{\ast }$.     

Global symmetric classical solutions of the full compressible Navier–Stokes equations with vacuum and large initial data

In this paper, we get a result on global existence of classical and strong solutions of the full compressible Navier–Stokes equations in three space dimensions with spherically or cylindrically symmetric initial data which may be large. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate a priori   estimates globally in time which depend on the assumption κ=O(1+θ^q)$\kappa =O(1+{\theta }^q)$ where q>r$q>r$ (r   can be zero), which relaxes the condition q?2+2r$q?2+2r$ in ,  and . This could be viewed as an extensive work of [16] where the equations hold in the sense of distributions in the set where the density is positive with initial data which is large, discontinuous, and spherically or cylindrically symmetric in three space dimension.     

Asymptotic behavior of the solution to the non-isothermal phase field equation

We consider the Penrose–Fife phase field model [Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62] with homogeneous Neumann boundary condition to the nonlinear heat flux q=∇(1/θ)$\mathbf{q}=\nabla (1/\theta )$, i.e., q=0$\mathbf{q}=0$ on the boundary, where θ>0$\theta >0$ is the temperature. There is a unique H¹$H^1$ solution globally in time with the non-empty, connected, compact ω$\omega$-limit set composed of stationary solutions, and the linearized stable stationary solution is dynamically stable.     

R-K type Landweber method for nonlinear ill-posed problems

In this paper we propose the R-K type Landweber iteration and investigate the convergence of the method for nonlinear ill-posed problem F(x)=y$F(x)=y$ where F:H→H$F:H\to H$ is a nonlinear operator between Hilbert space H  . Moreover, for perturbed data with noise level δ$\delta$ we prove that the convergence rate is O(δ^2/3)$\mathrm{O}({\delta }^{2/3})$ under appropriate conditions. Finally, the numerical performance of this R-K type Landweber iteration for a nonlinear convolution equation is compared with the Landweber iteration.     

Existence and uniqueness of periodic solutions for a p-Laplacian Duffing equation with a deviating argument

By means of Mawhin’s continuation theorem, we study a class of p-Laplacian Duffing type differential equations of the form

(_φp(x^′(t)))^′=Cx^′(t)+g(t,x(t),x(t−τ(t)))+e(t).${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }=C{x}^{\prime }\left(t\right)+g(t,x\left(t\right),x(t-\tau \left(t\right)))+e\left(t\right)\text{.}$

Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions

This paper investigates the solvability of discrete Dirichlet boundary value problems by the lower and upper solution method. Here, the second-order difference equation with a nonlinear right hand side f$f$ is studied and f(t,u,v)$f(t,u,v)$ can have a superlinear growth both in u$u$ and in v$v$. Moreover, the growth conditions on f$f$ are one-sided. We compute a priori bounds on solutions to the discrete problem and then obtain the existence of at least one solution. It is shown that solutions of the discrete problem will converge to solutions of ordinary differential equations.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

New results of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T$T$-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(t,x^′(t))+g(t,x(t))=e(t).$x^{″}+f(t,x^{\prime }\left(t\right))+g(t,x\left(t\right))=e\left(t\right)\text{.}$

Existence of periodic solutions for a Liénard type p-Laplacian differential equation with a deviating argument

By means of Mawhin’s continuation theorem, a class of p-Laplacian type differential equation with a deviating argument of the form

(_φp(x^′(t)))^′+f(x(t))x^′(t)+β(t)g(t,x(t−τ(t,_|x|∞)))=e(t)${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(x\left(t\right)\right)x^{\prime }\left(t\right)+\beta \left(t\right)g(t,x(t-\tau (t,{\left|x\right|}_{\infty })))=e\left(t\right)$

Periodic solutions to a second order nonlinear neutral functional differential equation in the critical case

In this paper, the authors study the existence of periodic solutions for a second order neutral functional differential equation

(x(t)-cx(t-τ))^″=f(x(t))x^′(t)+g(t,x(t-μ(t)))+e(t)$(x(t)-\mathit{cx}(t-\tau ){)}^{″}=f(x(t))x^{\prime }(t)+g(t,x(t-\mu (t)))+e(t)$