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## An Initial-Boundary Value Problem for the Korteweg–de Vries Equation with Dominant Surface Tension

### Acta Applicandae Mathematicae (2014-02-01) 129: 41-59 , February 01, 2014

We consider the initial-boundary value problem (IBVP) for the Korteweg–de Vries equation with zero boundary conditions at *x*=0 and arbitrary smooth decreasing initial data. We prove that the solution of this IBVP can be found by solving two linear inverse scattering problems (SPs) on two different spectral planes. The first SP is associated with the KdV equation. The second SP is self-conjugate and its scattering function is found in terms of entries of the scattering matrix *s*(*k*) for the first SP. Knowing the scattering function, we solve the second inverse SP for finding the potential self-conjugate matrix. Consequently, the unknown object entering coefficients in the system of evolution equations for *s*(*k*,*t*) is found. Then, the time-dependent scattering matrix *s*(*k*,*t*) is expressed in terms of *s*(*k*)=*s*(*k*,0) and of solutions of the self-conjugate SP. Knowing *s*(*k*,*t*), we find the solution of the IBVP in terms of the solution of the Gelfand–Levitan–Marchenko equation in the first inverse SP.

## Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid

### Acta Applicandae Mathematicae (1994-11-01) 37: 109-128 , November 01, 1994

The global existence of measure-valued solutions of initial boundary-value problems in bounded domains to systems of partial differential equations for viscous non-Newtonian isothermal compressible monopolar fluid and the global existence of the weak solution for multipolar fluid is proved.

## The Stability of the Quartic Functional Equation in Random Normed Spaces

### Acta Applicandae Mathematicae (2010-05-01) 110: 797-803 , May 01, 2010

The main problem analyzed in this paper consists in showing that, under some conditions, every almost quartic mapping from a linear space to a random normed space under the Łukasiewicz t-norm can be suitably approximated by a quartic function, which is unique.

## Global Weak Solutions of 3D Compressible Nematic Liquid Crystal Flows with Discontinuous Initial Data and Vacuum

### Acta Applicandae Mathematicae (2016-04-01) 142: 149-171 , April 01, 2016

In this paper, we study the global existence of weak solutions to the Cauchy problem of the three-dimensional equations for compressible isentropic nematic liquid crystal flows subject to discontinuous initial data. It is assumed here that the initial energy is suitably small in *L*^{2}, and the initial density, the gradients of initial velocity/liquid crystal director field are bounded in *L*^{∞}, *L*^{2} and *H*^{1}, respectively. This particularly implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial energy.

## Some recent results in finitely additive white noise theory

### Acta Applicandae Mathematicae (1994-05-01) 35: 27-47 , May 01, 1994

We present a short survey of some very recent results on the finitely additive white noise theory. We discuss the Markov property of the solution of a stochastic differential equation driven directly by a white noise, study the Radon-Nikodym derivative of the measure induced by nonlinear transformation on a Hilbert space with respect to the canonical Gauss measure thereon and obtain a representation for nonlinear filter maps.

## Compatibility, Multi-brackets and Integrability of Systems of PDEs

### Acta Applicandae Mathematicae (2010-01-01) 109: 151-196 , January 01, 2010

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer *δ*-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are used to establish new integration methods.

## The WDVV Equations in Pure Seiberg–Witten Theory

### Acta Applicandae Mathematicae (2005-03-01) 86: 49-102 , March 01, 2005

We review the relationship between pure four-dimensional Seiberg–Witten theory and the periodic Toda chain. We discuss the definition of the prepotential and give two proofs that it satisfies the generalized Witten–Dijkgraaf–Verlinde–Verlinde equations. A number of steps in the definitions and proofs that is missing in the literature is supplied.

## Existence of Solutions for Semilinear Nonlocal Elliptic Problems via a Bolzano Theorem

### Acta Applicandae Mathematicae (2013-10-01) 127: 87-104 , October 01, 2013

In this paper we deal with the existence of positive solutions for the following nonlocal type of problems
$$\everymath{\displaystyle} \left\{ \begin{array}{l@{\quad}l} -\Delta u = \frac{\sigma}{( \int_{\varOmega} g(u)\, dx )^p} f(u) & \mbox{in}\ \varOmega, \\[3mm] u>0 & \mbox{in}\ \varOmega, \\[1mm] u=0 & \mbox{on}\ \partial\varOmega, \end{array} \right. $$
where *Ω* is a bounded smooth domain in ℝ^{N} (*N*≥1), *f*,*g* are continuous positive functions, *σ*>0 and *p*∈ℝ.

We give sufficient conditions on the functions *f* and *g* in order to have existence of positive solutions.

## On Mixed cs-Groups

### Acta Applicandae Mathematicae (2005-01-01) 85: 75-80 , January 01, 2005

Description of mixed Abelian groups in which the closure of any pure subgroup in the *Z*-adic and *p*-adic topologies for every prime *p* is a direct summand of the initial group is obtained.