Periodic solutions of nonlinear functional difference equations at nonresonance case

Sufficient conditions for the existence of at least one T-periodic solution of nonlinear functional difference equation

Δx(n)+a(n)x(n)=f(n,u(n)),     

Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials

Let u be the weak solution to the degenerate Schrödinger equation with singular coefficients in Lipschitz domain as following

−div(w(x)A(x)∇u(x))+V(x)u(x)w(x)=0,     

A nonhomogeneous elliptic problem involving critical growth in dimension two

In this paper we study a class of nonhomogeneous Schrödinger equations

−Δu+V(x)u=f(u)+h(x)     

On a nonlinear eigenvalue problem in ODE

In this paper we shall study the following variant of the logistic equation with diffusion:

−du^″(x)=g(x)u(x)−u²(x)     

On the two-dimensional stationary Schrödinger equation with a singular potential

We study the equation

−△u(x,y)+ν(x,y)u(x,y)=0     

On a singular minimizing problem

We investigate the non-homogeneous modular Dirichlet problem Δ _p (·)u(x) = f (x) (where Δ _p (·)u(x) = div(|?u|^p(x-2)?u(x)) from the functional analytic point of view and we prove the stability of the solutions \({\left( {{u_{{p_i}}}} \right)_i}\) of the equation \({\Delta _{{p_i}\left( \cdot \right)}}{u_{{p_i}\left( \cdot \right)}} = f\) as p _i (·) → q(·) via Gamma-convergence of sequence of appropriate functionals.     

On the complete group classification of the reaction-diffusion equation with a delay

The reaction-diffusion delay differential equation

ut(x,t)−uxx(x,t)=g(x,u(x,t),u(x,t−τ))     

A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation

$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$

on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u₀(x) = u(x, 0) and the boundary data g₀(t) = u(0, t), g₁(t) = u_x(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

     

New results of general n-dimensional incompressible Navier-Stokes equations

Let u=u(x,t,u₀) represent the global strong/weak solutions of the Cauchy problems for the general n-dimensional incompressible Navier-Stokes equations

     

An inverse problem of identifying the coefficient in a nonlinear parabolic equation

This paper deals with the determination of a pair (p,u) in the nonlinear parabolic equation

u_t−u_xx+p(x)f(u)=0,     

New exact general solutions of Abel equation of the second kind

The Abel equation of the second kind

[g₀(x)+g₁(x)u]u^′=f₀(x)+f₁(x)u+f₂(x)u²     

Solutions of second-order and fourth-order ODEs on the half-line

We start by studying the existence of positive solutions for the differential equation

u^″=a(x)u−g(u),     

On power series representing solutions of the one-dimensional time-independent Schrödinger equation

For the equation χ″(x) = u(x)χ(x) with infinitely smooth u(x), the general solution χ(x) is found in the form of a power series. The coefficients of the series are expressed via all derivatives u ^(m)(y) of the function u(x) at a fixed point y. Examples of solutions for particular functions u(x) are considered.     

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.     

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

We consider random self-adjoint Jacobi matrices of the form

(Jωu)(n)=an(ω)u(n+1)+bn(ω)u(n)+an−1(ω)u(n−1)     

Global Hölder continuity of weak solutions to quasilinear divergence form elliptic equations

We derive global Hölder regularity for the -weak solutions to the quasilinear, uniformly elliptic equation

div(aij(x,u)Dju+ai(x,u))+a(x,u,Du)=0     

A tracing of the fractional temperature field

This paper is devoted to a study of L~q-tracing of the fractional temperature field u(t, x)—the weak solution of the fractional heat equation(?_t +(-?_x)~α)u(t, x) = g(t, x) in L~p(R_+~(1+n)) subject to the initial temperature u(0, x) = f(x) in L~p(R~n).     

Study of the solutions to a model porous medium equation with variable exponent of nonlinearity

The authors of this paper study the Dirichlet problem of the following equation

ut−div(|u|^ν(x,t)∇u)=f−|u|^p(x,t)−1u.     

Asymptotic spreading in heterogeneous diffusive excitable media

We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type:

t_∂u−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u)     

Radial ground states and singular ground states for a spatial-dependent p-Laplace equation

We consider the following equation

Δpu(x)+f(u,|x|)=0,