arXiv subjects
Publications and source records attributed to Madhumita Roy.
We establish a simple and explicit criterion for wave breaking for a general class of perturbed Burgers equations that cover several Burgers-type models, including the Fractional KdV equation, the Whitham equation, and the Fornberg-Whitham equation. The proof is both rigorous and straightforward.
This paper studies the existence of minimal solutions to two-point boundary value problems for quasi-monotone dynamical systems. Specifically, the pointwise infimum of all supersolutions is shown to coincide with the minimal solution. This result is then applied to establish a non-uniqueness result for strong stable solutions to a class of mean field games with a continuum of players.
A long-time behavior of solutions to a nonlinear plate model subject to non-conservative and non-dissipative effects and nonlinear damping is considered. The model under study is a prototype for a suspension bridge under the effects of unstable flow of gas. To counteract the unwanted oscillations a damping mechanism of a nonlinear nature is applied. From the point of view of nonlinear PDEs, we are dealing with a non-dissipative and nonlinear second order in time dynamical system of hyperbolic nature subjected to nonlinear damping. One of the first goals is to establish ultimate dissipativity of all solutions, which will imply an existence of a weak attractor. The combined effects of non-dissipative forcing with nonlinear damping-leading to an overdamping-give rise to major challenges in proving an existence of an absorbing set. Known methods based on equipartition of the energy do not suffice. A rather general novel methodology based on ``barrier's'' method will be developed to address this and related problems. Ultimately, it will be shown that a weak attractor becomes strong, and the nonlinear PDE system has a coherent finite-dimensional asymptotic behavior.
An increasing sequence $(a_n)$ of positive integers which satisfies $\frac{a_{n+1}}{a_n}>1+η$ for some positive $η$ is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence is strong sweeping out which means that in every aperiodic dynamical system we can find a set $E$ of arbitrary small measure so that $\limsup_N\frac{1}{N} \sum_{n\le N}\mathbb{1}_E(T^nx)=1$ and $\liminf_N\frac{1}{N} \sum_{n\le N}\mathbb{1}_E(T^nx)=0$ almost everywhere. In this paper we improve this result by showing that if $(a_n)$ satisfies only $\frac{a_{n+1}}{a_n}>1+\frac1{(\log\log n)^{1-η}}$ for some positive $η$ then it is already strong sweeping out.
One of the critical measures to control infectious diseases is a lockdown. Once past the lockdown stage in many parts of the world, the crucial question now concerns the effects of relaxing the lockdown and finding the best ways to implement further lockdown(s), if required, to control the spread. With the relaxation of lockdown, people migrate to different cities and enhance the spread of the disease. This work presents the population migration model for n-cities and applies the model for migration between two and three cities. The reproduction number is calculated, and the effect of the migration rate is analyzed. A punctuated lockdown is implemented to simulate a protocol of repeated lockdowns that limits the resurgence of infections. A damped oscillatory behavior is observed with multiple peaks over a period.
We present a model analysis of the tumor and normal cell growth under the influence of a carcinogenic agent, an immunomdulator (IM) and variable influx of immune cells including relevant interactions. The tumor growth is facilitated by carcinogens such as inorganic arsenic while the IM considered here is black tea (Camellia sinesnsis). The model with variable influx of immune cells is observed to have considerable advantage over the constant influx model, and while the tumor cell population is greatly mitigated, normal cell population remains above healthy levels. The evolution of normal and tumor cells are computed from the proposed model and their local stabilities are investigated analytically. Numerical simulations are performed to study the long term dynamics and an estimation of the effects of various factors is made. This helps in developing a balanced strategy for tumor mitigation without the use of chemotherapeutic drugs that usually have strong side-effects.
Chronic exposure to inorganic arsenic is a potential cause of carcinogenesis. It elicits its potential by generation of ROS, leading to DNA, protein and lipid damage. Therefore, the deleterious effect of arsenic can be mitigated by quenching ROS using antioxidants. There is a homology between the protein coding regions of mice and human. Effect of these alterations in human can be mimicked in mice. Therefore to understand the underlying mechanism of arsenic toxicity and its amelioration by black tea, studies have been conducted in mice model. Long term exposure to iAs leads to tumour growth, which has been found to be alleviated by black tea. Observations reveal that black tea has two salutary effects on the growth of tumour: the rate of growth of damaged cells was appreciably reduced and an early saturation of the level of damage is achieved. To take the experimental findings further, the experimental data have been modelled with simple dynamical equations. The curves obtained from \textit{in vivo} studies have been fitted with the data obtained from the model. The corresponding steady states and their stabilities are analyzed.