Search

We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger's type equations, convective Cahn-Hilliard equation, generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish the following common scenario: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similarly to the case when the equation does not involve convective term. This kind of result has been previously known for the case of Burger's type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.

The dissipative wave equation with a critical quintic non-linearity in smooth bounded three-dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.

We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity epsilon(l) and a complex Kerr coefficient sigma admits blow-up solutions, provided that the real part of a is negative, i.e., the slab is defocusing. This result applies to homogeneous as well as inhomogeneous Kerr slabs if epsilon(l) and sigma are continuous functions of the transverse coordinate, and the real part of sigma is bounded above by a negative number. It shows that a recently reported nonlinear optical amplification effect, which relies on the existence of blow-up solutions, persists the presence of losses and transverse inhomogeneities., Scientific and Technological Research Council of Turkey (TÜBİTAK); Turkish Academy of Sciences (TÜBA)

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation. This algorithm capitalizes on the fact that such infinite-dimensional dissipative dynamical systems posses finite-dimensional long-time behavior which is represented by, for instance, the finitely many determining parameters of their long-time dynamics, such as determining Fourier modes, determining volume elements, determining nodes , etc..The algorithm utilizes these finite parameters in the form of feedback control to stabilize the relevant solutions. For the sake of clarity, and in order to fix ideas, we focus in this work on the case of low Fourier modes feedback controller, however, our results and tools are equally valid for using other feedback controllers employing other spatial coarse mesh interpolants.

This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least -smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a -smooth inertial manifold may not exist. On the other hand, since an attractor usually has finite fractal dimension, by Mane's theorem it projects bijectively and Holder-homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mane projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness. Bibliography: 35 titles.

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with a nonlinearity of arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.

We consider the 2D Navier–Stokes–Voight equation in an unbounded strip-like domain. It is shown that the semigroup generated by this equation has a global attractor in weighted Sobolev spaces.

WOS:000260209700005 We consider the initial-boundary value problem for the double diffusive convective Brinkman equations with homogenous Neumann's boundary conditions. The continuous dependence of solutions of the given problem on the Soret constant in H-2(Omega) boolean AND H-0(1)(Omega) x H-1(Omega) x H-1(Omega) is proved.

A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved, the initial boundary value problem for the considered equation is globally well-posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function. (C) 2015 AIP Publishing LLC, Scientific and Technological Research Council of Turkey (TÜBİTAK)

Exploring autonomic nervous system (ANS) changes during hypnosis is critical for understanding the nature and extent of the hypnotic phenomenon and for identifying the mechanisms underlying the effects of hypnosis in different medical conditions. To assess ANS changes during hypnosis, electrodermal activity and pulse rate variability (PRV) were measured in 121 young adults. Participants either received hypnotic induction (hypnosis condition) or listened to music (control condition), and both groups were exposed to test suggestions. Blocks of silence and experimental sound stimuli were presented at baseline, after induction, and after de-induction. Skin conductance level (SCL) and high frequency (HF) power of PRV measured at each phase were compared between groups. Hypnosis decreased SCL compared to the control condition; however, there were no group differences in HF power. Furthermore, hypnotic suggestibility did not moderate ANS changes in the hypnosis group. These findings indicate that hypnosis reduces tonic sympathetic nervous system activity, which might explain why hypnosis is effective in the treatment of disorders with strong sympathetic nervous system involvement, such as rheumatoid arthritis, hot flashes, hypertension, and chronic pain. Further studies with different control conditions are required to examine the specificity of the sympathetic effects of hypnosis.

This paper is concerned with global in time behavior of solutions for a semilinear, hyperbolic, inverse source problem. We prove two types of results. The first one is a global nonexistence result for smooth solutions when the data is chosen appropriately. The second type of results is the asymptotic stability of solutions when the integral constraint vanishes as t goes to infinity. Bibliography: 22 titles.

Author(s): Kalantarov, VK; Titi, ES | Abstract: In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of various classes of damped nonlinear wave equations. Specifically, stabilization the zero steady state solution of initial boundary value problems for nonlinear weakly and strongly damped wave equations, nonlinear wave equation with nonlinear damping term and some related nonlinear wave equations, introducing a feedback control terms that employ parameters, such as, finitely many Fourier modes, finitely many volume elements and finitely many nodal observables and controllers. In addition, we also establish the stabilization of the zero steady state solution to initial boundary value problem for the damped nonlinear wave equation with a controller acting in a proper subdomain. Notably, the feedback controllers proposed here can be equally applied for stabilizing other solutions of the underlying equations.

We show that solutions of the initial boundary value problem for the coupled system of Kuramoto-Sivashinsky and Ginzburg-Landau (KS–GL) equations continuously depend on parameters of the system, and under some restrictions on parameters all solutions of initial boundary value problem for KS–GL equations tend to zero as with an exponential rate.

Our aim is to establish a spatial decay and growth estimates for solutions of the initial-boundary value problem for the linear wave equation with the damping term under nonlinear boundary conditions.

We consider the periodic initial-boundary value problem for a multidimensional generalized Benjamin–Bona–Mahony equation. We show the existence of the global attractor with a finite fractal dimension and the existence of the exponential attractor for the corresponding semigroup.