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A general system of several ordinary differential equations coupled with a reaction-diffusion equation in a bounded domain with zero-flux boundary condition is studied in the context of pattern formation. These initial-boundary value…
We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form $$ \begin{cases} \displaystyle u_t -…
The well-known solution theory for (systems of) linear ordinary differential equations undergoes significant changes when introducing an additional real parameter. Properties like the existence of fundamental sets of solutions or…
We introduce two stationary versions of two discrete variants of Hammersley's process in a finite box, this allows us to recover in a unified and simple way the laws of large numbers proved by T. Sepp{\"a}l{\"a}inen for two generalized…
We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity…
Given a probability measure $\mu$ with infinite support on the unit circle $\partial\mathbb{D}=\{z:|z|=1\}$, we consider a sequence of paraorthogonal polynomials $\h_n(z,\lambda)$ vanishing at $z=\lambda$ where $\lambda \in \T$ is fixed. We…
A wide class of non-autonomous nonlinear parabolic partial differential equations with delay is studied. We allow in our investigations different types of delays such as constant, time-dependent, state-dependent (both discrete and…
A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter.…
In this work we investigate the existence of solutions, their uniqueness and finally dependence on parameters for solutions of second order neutral nonlinear difference equations. The main tool which we apply is Darbo fixed point theorem.
In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation: $$-\Delta u-\lambda u=\left(\frac{1}{|x|^{\mu}}\ast A|u|^{p}\right)A|u|^{p-2}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c,\quad u\in…
We examine the fourth order problem $\Delta^2 u = \lambda f(u) $ in $ \Omega$ with $ \Delta u = u =0 $ on $ \partial \Omega$, where $ \lambda > 0$ is a parameter, $ \Omega$ is a bounded domain in $ R^N$ and where $f$ is one of the following…
This paper develops and analyzes an accelerated proximal descent method for finding stationary points of nonconvex composite optimization problems. The objective function is of the form $f+h$ where $h$ is a proper closed convex function,…
We investigate the point singularity of very weak solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations. More precisely, assume that the solution $(\mathbf{u},\mathbf{B})$ in the punctured ball $B_2\setminus \{0\}$ satisfies…
The article continues the work on the description of integrable nonlinear chains with three independent variables of the following form $u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n}, u^{j}_n,u^j_{n+1 },u^{j-1}_{n+1})$ by the presence of a hierarchy…
In this paper we are concerned with the existence of ground states solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} -_tD^\alpha_\infty(_{-\infty}D^\alpha_t u(t)) - \lambda L(t)u(t)+\nabla…
The solution $x_n\left(t\right)$, $n=1,2,$ of the \textit{initial-values} problem is reported of the \textit{autonomous} system of $2$ coupled first-order ODEs with \textit{homogeneous cubic polynomial} right-hand sides, \begin{eqnarray}…
We study the Schr\"{o}dinger-Poisson type system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\left( \mu _{11}\phi _{u}-\mu _{12}\phi _{v}\right) u=% \frac{1}{2\pi }\int_{0}^{2\pi }\left\vert u+e^{i\theta }v\right\vert…
In this paper, we determine a concrete interval of positive parameters $\lambda$, for which we prove the existence of infinitely many homoclinic solutions for a discrete problem $-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right)…
We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions…
A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the…