Non-homogeneous random walks, subdiffusive migration of cells and anomalous chemotaxis

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.Comment: 18 pages, accepted for publicatio

Development of singularities for the compressible Euler equations with external force in several dimensions

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem by Sideris on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.Comment: 23 page

A time discretization scheme for a nonlocal degenerate problem modelling resistance spot welding

This is the author's PDF version of an article published in Mathematical Modelling of Natural Phenomena© 2015. The definitive version is available at http://www.mmnp-journal.org/articles/mmnp/abs/2015/06/mmnp2015106p90/mmnp2015106p90.htmlIn the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm

Dynamical extensions for shell-crossing singularities

We derive global weak solutions of Einstein's equations for spherically symmetric dust-filled space-times which admit shell-crossing singularities. In the marginally bound case, the solutions are weak solutions of a conservation law. In the non-marginally bound case, the equations are solved in a generalized sense involving metric functions of bounded variation. The solutions are not unique to the future of the shell-crossing singularity, which is replaced by a shock wave in the present treatment; the metric is bounded but not continuous.Comment: 14 pages, 1 figur

The Speed of Fronts of the Reaction Diffusion Equation

We study the speed of propagation of fronts for the scalar reaction-diffusion equation $u_t = u_{xx} + f(u)$\, with $f(0) = f(1) = 0$. We give a new integral variational principle for the speed of the fronts joining the state $u=1$ to $u=0$. No assumptions are made on the reaction term $f(u)$ other than those needed to guarantee the existence of the front. Therefore our results apply to the classical case $f > 0$ in $(0,1)$, to the bistable case and to cases in which $f$ has more than one internal zero in $(0,1)$.Comment: 7 pages Revtex, 1 figure not include

Thrombospondin 2 expression is correlated with inhibition of angiogenesis and metastasis of colon cancer

Two subtypes of thrombospondin (TSP-1 and TSP-2) have inhibitory roles in angiogenesis in vitro, although the biological significance of these TSP isoforms has not been determined in vivo. We examined TSP-1 and TSP-2 gene expression by reverse transcription polymerase chain reaction (RT-PCR) analysis in 61 colon cancers. Thirty-eight of these 61 colon cancers were positive for TSP-2 expression and showed hepatic metastasis at a significantly lower incidence than those without TSP-2 expression (P = 0.02). TSP-2 expression was significantly associated with M0 stage in these colon cancers (P = 0.03), whereas TSP-1 expression showed no apparent correlation with these factors. The colon cancer patients with TSP-2 expression showed a significantly low frequency of liver metastasis correlated with the cell-associated isoform of vascular endothelial growth factor (VEGF-189) (P = 0.0006). Vascularity was estimated by CD34 staining, and TSP-2(–)/VEGF-189(+) colon cancers showed significantly increased vessel counts and density in the stroma (P < 0.0001). TSP-2(–)/VEGF-189(+) colon cancer patients also showed significantly poorer prognosis compared with those with TSP-2(+) / VEGF-189(–) (P = 0.0014). These results suggest that colon cancer metastasis is critically determined by angiogenesis resulting from the balance between the angioinhibitory factor TSP-2 and angiogenic factor VEGF-189. © 1999 Cancer Research Campaig

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

Angiostatin generating capacity and anti-tumour effects of D-penicillamine and plasminogen activators

BACKGROUND: Upregulation of endogenous angiostatin levels may constitute a novel anti-angiogenic, and therefore anti-tumor therapy. In vitro, angiostatin generation is a two-step process, starting with the conversion of plasminogen to plasmin by plasminogen activators (PAs). Next, plasmin excises angiostatin from other plasmin molecules, a process requiring a donor of a free sulfhydryl group. In previous studies, it has been demonstrated that administration of PA in combination with the free sulfhydryl donor (FSD) agents captopril or N-acetyl cysteine, resulted in angiostatin generation, and anti-angiogenic and anti-tumour activity in murine models. METHODS: In this study we have investigated the angiostatin generating capacities of several FSDs. D-penicillamine proved to be most efficient in supporting the conversion of plasminogen to angiostatin in vitro. Next, from the optimal concentrations of tPA and D-penicillamine in vitro, equivalent dosages were administered to healthy Balb/c mice to explore upregulation of circulating angiostatin levels. Finally, anti-tumor effects of treatment with tPA and D-penicillamine were determined in a human melanoma xenograft model. RESULTS: Surprisingly, we found that despite the superior angiostatin generating capacity of D-penicillamine in vitro, both in vivo angiostatin generation and anti-tumour effects of tPA/D-penicillamine treatment were impaired compared to our previous studies with tPA and captopril. CONCLUSION: Our results indicate that selecting the most appropriate free sulfhydryl donor for anti-angiogenic therapy in a (pre)clinical setting should be performed by in vivo rather than by in vitro studies. We conclude that D-penicillamine is not suitable for this type of therapy

Front-like entire solutions for monostable reaction-diffusion systems

This paper is concerned with front-like entire solutions for monostable reactiondiffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Combining a SIS and traveling fronts with different wave speeds and directions, the existence and various qualitative properties of entire solutions are then established using comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish some comparison arguments for the three systems. The existence of entire solutions is then proved via the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems