Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems

We analyse the dynamics of the non-autonomous nonlinear reaction-diffusion equation

ut−Δu=f(t,x,u),     

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     

Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation

ut−diva(x,∇u)+f(x,u)=0     

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))     

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     

Infinitely many periodic solutions for a class of second-order Hamiltonian systems

In this paper we study the existence of infinitely many periodic solutions for second-order Hamiltonian systems

$$\left\{ {\begin{array}{*{20}c} {\ddot u(t) + A(t)u(t) + \nabla F(t,u(t)) = 0,} \\ {u(0) - u(T) = \dot u(0) - \dot u(T) = 0,} \\ \end{array} } \right.$$

, where F(t, u) is even in u, and ?F(t, u) is of sublinear growth at infinity and satisfies the Ahmad-Lazer-Paul condition.

     

Optimal conditions of solvability of nonlocal problems for second-order ordinary differential equations

For the differential equation

u^″=f(t,u)     

Nonhomogeneous boundary value problems for some nonlinear equations with singular ?-Laplacian

Using Leray-Schauder degree theory we obtain various existence results for the quasilinear equation problems

(?(u^′)^)′=f(t,u,u^′)     

Boundary estimates for solutions to operators of p-Laplace type with lower order terms

In this paper we study the boundary behavior of solutions to equations of the form

∇⋅A(x,∇u)+B(x,∇u)=0,     

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.

     

Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control

In this work we provide an existence and location result for the third-order nonlinear differential equation

u^?(t)=f(t,u(t),u^′(t),u^″(t)),     

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),     

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.     

Gevrey micro-regularity for solutions to first order nonlinear PDE

Let (x,t)∈Rm×R and u∈C²(Rm×R). We study the Gevrey micro-regularity of solutions u of the nonlinear equation

ut=f(x,t,u,ux),     

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

     

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)     

Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation

We consider the Cauchy problem for the nonlinear differential equation

$$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$

where ? > 0 is a small parameter, f(x, u) ∈ C ^∞ ([0, d] × ?), R ₀ > 0, and the following conditions are satisfied: f(x, u) = x ? u ^p + O(x ² + |xu| + |u|^p+1) as x, u → 0, where p ∈ ? \ {1} f(x, 0) > 0 for x > 0; f _u ²(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ ₀ ^+∞ f _u ²(x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].

     

On some third order nonlinear boundary value problems: Existence, location and multiplicity results

We prove an Ambrosetti-Prodi type result for the third order fully nonlinear equation

u^?(t)+f(t,u(t),u^′(t),u^″(t))=sp(t)     

On the Fu?ik spectrum of the wave operator and an asymptotically linear problem

We study generalized solutions of the nonlinear wave equation

utt−uss=au⁺−bu⁻+p(s,t,u),     

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,