Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

Large-scale Marangoni convection in a liquid layer with insoluble surfactant of low concentration

We derive a system of amplitude equations describing the evolution of a large-scale Marangoni patterns in a liquid layer with poorly conducting boundaries in the presence of a small amount of an insoluble surfactant on the free flat interface. The presence of quadratic nonlinear terms in the amplitude equation leads to the selection of hexagonal patterns. The type of hexagons bifurcating into the subcritical region, depends on the parameters of the system

On dynamic excitation of Marangoni instability in a liquid layer with insoluble surfactant on the deformable surface

This paper is a continuation of our previous work presented at the IMA-6, see [1]. We continue to analyze the parametric excitation of Marangoni instability by a periodic flux modulation in a liquid layer with insoluble surfactant. Contrary to the previous investigation here the upper free surface of the layer is deformable. The linear stability analysis for the disturbances with arbitrary wave numbers is performed. Three response modes of the system to an external periodic stimulation were found – synchronous, subharmonic, and quasi-periodic ones. Results for different Galileo and inverse capillary parameters are presented. It is shown that contrary to the situation with nondeformable interface, at small values of Galileo and inverse capillary parameters a new subharmonic instability region appears in the range of long waves

Marangoni Patterns in a Non-Isothermal Liquid with Deformable Interface Covered by Insoluble Surfactant

Marangoni patterns are created by instabilities caused by thermocapillary and solutocapillary stresses on the deformable free surface of a thin liquid layer. In the present paper, we consider the influence of the insoluble surfactant on the selection and modulational instability of stationary Marangoni patterns near their onset threshold. The basic governing parameters of the problem are the Biot number characterizing the heat-transfer resistances of and at the surface, the Galileo number indicating the role of gravity via viscous forces, and the elasticity number specifying the influence of insoluble surfactant on the interfacial dynamics of the liquid. The paper includes a review of the previous results obtained in that problem as well as new ones

Parametric excitation of oscillatory Marangoni instability in a heated liquid layer covered by insoluble surfactant

This paper is a continuation of our previous research reported in the two last IMA conferences. While in our previous works the driving modulation was a modulation of the temperature gradient, here the temperature gradient is fixed in the whole liquid layer and the liquid is vertically vibrated with some fixed driving frequency. Two kinds of waves can be generated in this system. The first kind of waves are transverse (capillary-gravity) waves. The second kind of waves are longitudinal (Marangoni) waves caused by the dependence of the surface tension on the temperature and the surfactant concentration. A linear analysis with arbitrary wave numbers is performed. Two different heating regimes were considered. Multiple instability regions depending on the heating conditions are analyzed. Among three possible modes of the system's response to external forcing, the most “dangerous” one is the subharmonic instability mode

Patterns and Their Large-Scale Distortions in Marangoni Convection with Insoluble Surfactant

Nonlinear dynamics of patterns near the threshold of long-wave monotonic Marangoni instability of conductive state in a heated thin layer of liquid covered by insoluble surfactant is considered. Pattern selection between roll and square planforms is analyzed. The dependence of pattern stability on the heat transfer from the free surface of the liquid characterized by Biot number and the gravity described by Galileo number at different surfactant concentrations is studied. Using weakly nonlinear analysis, we derive a set of amplitude equations governing the large-scale roll distortions in the presence of the surface deformation and the surfactant redistribution. These equations are used for the linear analysis of modulational instability of stationary rolls