We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -\Delta + m^2} \ u - mu V(x)u \frac{\mu}{|x|} = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-\alpha}} \, dy \right) f(x,u) K (x) |u|^{q-2}u \end{multline*} under suitable assumptions on bounded potential \(V\) and nonlinearity \(f\). Our analysis extends recent results by second third author problem with $...

The current article studies optical solitons solutions for the dimensionless form of stochastic resonant nonlinear Schrödinger equation (NLSE) with both spatio-temporal dispersion (STD) and inter-modal (IMD) having multiplicative noise in itô sense. We will discuss seven laws nonlinearities, namely, Kerr law, power parabolic dual-power quadratic–cubic polynomial triple-power law. new auxiliary ...

This paper deals with modelling and parameter identification of nonlinear systems described by Hammerstein model having asymmetric static nonlinearities known as preload nonlinearity characteristic. The simultaneous use of both an easy decomposition technique and the generalized orthonormal bases leads to a particular form of Hammerstein model containing a minimal parameters number. The employ ...

The study of differential equations and variational problems with nonstandard p(x)growth conditions is a new and interesting topic. We refer to [1, 2], the background of these problems. Many results have been obtained on this kind of problems, for example, [2–13]. In [4, 7], Fan and Zhao give the regularity of weak solutions for differential equations with nonstandard p(x)-growth conditions. On...

When modeling chemical processes we are concerned with mathematical models of more and more complexity: it is especially the coupling of equations describing different physical phenomena which adds a new level of complexity to the already present difficulties (3D domains, singular perturbations, heavy nonlinearities). In turn, this gives rise to an intolerable large computation time. In the pre...

0 gi(t, s)[Pi(s, u1(s), u2(s), · · · , un(s)) + Qi(s, u1(s), u2(s), · · · , un(s))]ds, t ∈ [0, T ], 1 ≤ i ≤ n where T > 0 is fixed and the nonlinearities Pi(t, u1, u2, · · · , un) can be singular at t = 0 and uj = 0 where j ∈ {1, 2, · · · , n}. Criteria are offered for the existence of fixed-sign solutions (u∗1, u ∗ 2, · · · , u ∗ n) to the system of Volterra integral equations, i.e., θiu ∗ i (...

where φp(s) is a p-Laplacian operator, that is, φp(s)= |s|p−2s, p > 1, η ∈ (0,1) is a given constant, α > 0, γ > 0, β ≥ 0, δ ≥ 0, g : [0,1]→ [0,∞) has countable many singularities on (0,1/2). In recent years, because of the wide mathematical and physical backgrounds [7, 8], the existence of positive solutions for nonlinear boundary value problems with p-Laplacian received wide attention. Especi...

We give the regularization for fractional integral by delta sequence and apply it to obtain existence-uniqueness theorems in Colombeau algebras for nonlinear equations with singularities: nonlinear system of integral equations with polar kernel and nonlinear parabolic equations (of ordinary type, with nonlinear conservative term and with Schrödinger kernel) with strongly singular initial data a...

We discuss the asymptotic behavior of solutions to a higher-order Emden–Fowler type equation with constant potential. Several author’s results are presented concerning both positive and oscillatory solutions to equations with regular and singular nonlinearities. We discuss the existence and asymptotic behavior of “blowup” solutions. Results on the asymptotic behavior of oscillating solutions ar...

A class of nonlinear parabolic systems with quadratic nonlinearities in the gradient (the case of two spatial variables) is considered. It is assumed that the elliptic operator of the system has a variational structure. The behavior of a smooth on a time interval [0, T ) solution to the Cauchy-Neumann problem is studied. For the situation when the “local energies” of the solution are uniformly ...