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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.

## 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.

## 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.

## 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.

## Periodic Solutions of Second Order Superlinear Singular Dynamical Systems

### Acta Applicandae Mathematicae (2010-08-01) 111: 179-187 , August 01, 2010

We study the existence of periodic solutions of second order superlinear dynamical systems with a singularity of repulsive type. The proof is based on a well-known fixed point theorem for completely continuous operators. We do not need to consider so-called strong force conditions. Recent results in the literature are generalized and significantly improved.

## Strong Solutions to Non-stationary Channel Flows of Heat-Conducting Viscous Incompressible Fluids with Dissipative Heating

### Acta Applicandae Mathematicae (2011-12-01) 116: 237-254 , December 01, 2011

We study an initial-boundary-value problem for time-dependent flows of heat-conducting viscous incompressible fluids in channel-like domains on a time interval (0,*T*). For the parabolic system with strong nonlinearities and including the artificial (the so called “do nothing”) boundary conditions, we prove the local in time existence, global uniqueness and smoothness of the solution on a time interval (0,*T*^{∗}), where 0<*T*^{∗}≤*T*.

## A Dynamical Tikhonov Regularization for Solving Ill-posed Linear Algebraic Systems

### Acta Applicandae Mathematicae (2013-02-01) 123: 285-307 , February 01, 2013

The Tikhonov method is a famous technique for regularizing ill-posed linear problems, wherein a regularization parameter needs to be determined. This article, based on an invariant-manifold method, presents an adaptive Tikhonov method to solve ill-posed linear algebraic problems. The new method consists in building a numerical minimizing vector sequence that remains on an invariant manifold, and then the Tikhonov parameter can be optimally computed at each iteration by minimizing a proper merit function. In the *optimal vector method* (OVM) three concepts of optimal vector, slow manifold and Hopf bifurcation are introduced. Numerical illustrations on well known ill-posed linear problems point out the computational efficiency and accuracy of the present OVM as compared with classical ones.

## Gap Solitons in Periodic Discrete Schrödinger Equations with Nonlinearity

### Acta Applicandae Mathematicae (2010-03-01) 109: 1065-1075 , March 01, 2010

By using the critical point theory, the existence of gap solitons for periodic discrete nonlinear Schrödinger equations is obtained. An open problem proposed by Professor Alexander Pankov is solved.

## On the Prandtl Boundary Layer Equations in Presence of Corner Singularities

### Acta Applicandae Mathematicae (2014-08-01) 132: 139-149 , August 01, 2014

In this paper we prove the well-posedness of the Prandtl boundary layer equations on a periodic strip when the initial and the boundary data are not assigned to be compatible.