A quantization property for blow up solutions of singular Liouville-type equations

Motivated by the study of self-dual vortices in gauge field theory (cf. (Vortices and Monopoles, Birkhauser, Boston, 1980; Solitons in Field Theory and Nonlinear Analysis, Monographs in Mathematics, Springer, New York, 2001)), we consider a “concentrating”solution-sequence u_k satisfying

     

非局部抛物型方程解的性质

In this short note, we investigate the properties of positive solutions for some non-local parabolic equations. The conditions on the global existence and blow-up in finite time of solution are given.     

PVDF/TiO2 nanocomposites prepared by solution blow spinning: Surface properties and their relation with S. Mutans adhesion

Thermoplastic nanocomposite materials with potential bactericide properties were prepared and their surface properties and adhesion to Streptococcus mutans, S. mutans, were characterized. Solution blow spinning was successfully used to prepare films with a fiber-like structure on the surface of nanocomposites based on Polyvinylidene fluoride, PVDF, filled with well dispersed TiO₂ nanoparticles. PVDF/TiO₂ nanocomposites were prepared varying the nanoparticles content (0%, 1%, 2%, 5% and 10% by weight). In order to understand the influence of the presence of TiO₂ nanoparticles and the final surface properties on the adhesion of S. mutans to the materials, a deep characterization was carried out focusing on the morphology, roughness, surface free energy from contact angle measurements and cell adhesion by single cell force spectroscopy. It was observed that the uniform dispersion of the nanofiller arose from nanoparticles embedded in the polymer when fibers were created during the blow spinning process. TiO₂ content influenced the topography of the films probably due to a direct effect on the solvent evaporation rate. Although this factor greatly conditioned the roughness of the samples and therefore the surface free energy, S. mutant adhesion on the substrates under study was concluded to be more dependent on the specific interactions with the surface polar groups of the material.     

Two blowup solutions for the inhomogeneous isotropic Landau–Lifshitz equation

Two exact blowup solutions of the (2+1)$(2+1)$-dimensional space–time inhomogeneous isotropic Landau–Lifshitz equation (IILL for short) are constructed under suitable transformations. These blowup examples show that some solutions of the 2-dimensional IILL on S²$S^2$ can develop a finite time singularity from smooth initial data with finite energy in finite (or infinite) spatial domain. Some properties of one of these solutions are illustrated in plots.     

On a nonlinear heat equation with viscoelastic term associated with Robin conditions

This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions.     

Life span of positive solutions for a semilinear heat equation with general non-decaying initial data

We prove upper bounds on the life span of positive solutions for a semilinear heat equation. For non-decaying initial data, it is well known that the solutions blow up in finite time. We give two types estimates of the life span in terms of the limiting values of the initial data in space.     

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin–Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.     

LOCAL EXISTENCE AND BLOW UP IN NONLINEAR THERMOELASTICITY WITH SECOND SOUND

ABSTRACT

In this work we establish a local existence and a blow up result for a multidimensional nonlinear system of thermoelasticity with second sound.     

Some blowup solutions about two systems derived from Landau–Lifshitz–Gilbert equation

Two different systems which can be regard as the extreme cases of Landau-Lifshitz–Gilbert equation are considered. When the gyromagnetic term of Landau–Lifshitz–Gilbert equation vanishes away, we consider some special solutions for a modified harmonic map heat flow which map from (2+1)$(2+1)$-dimensional space–time into the 2-sphere. The existence of regular initial data leading to blow up in finite time is established. When the Gilbert term is omitted, a blowup solution is obtained by constructing a blowup solution of Landau–Lifshitz equation.