In this paper we study a nonlinear reaction-diffusion system which models an infectious disease caused by bacteria such as cholera. One of the features in this model is that a certain portion of the recovered human hosts lost a lifetime immunity and could be infected again. Another feature in the model is that the mobility for each species is assumed to be dependent upon location and time. We also assume that the whole group is susceptible with the bacteria. This leads to a strongly coupled nonlinear reaction-diffusion system. We prove that the nonlinear system has a unique solution globally in any space dimension under some natural conditions on known parameters and functions. Moreover, the long-time behavior and stability analysis for the solution are carried out rigorously. In particular, we characterize the precise conditions on variable parameters about the stability or instability for all steady-state solutions. These results obtained in this paper answered several open questions raised in the previous literature

In this paper, a necessary and sufficient condition for the stability of Lyapunov exponents of linear differential system is proved in the sense that the equations satisfy the weaker form of integral separation instead of its classical one. Furthermore, the existence of full nonuniform exponential dichotomy spectrum under the condition of weak integral separateness is also presented.

The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problem \[ \frac{d}{dx}(1+\delta y+\gamma y^{2})^{n}\frac{dy}{dx}]+2x\frac{dy}{dx}=0,\,\,\,x>0,\,\,y(0)=0,\,\,\,y(\infty)=1, \] which was proposed in 1974 by [1] to represent a Stefan problem with a nonlinear temperature-dependent thermal conductivity on the semi-infinite line (0;1). The modified error function of two parameters $\varphi_{\delta,\gamma}$ is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in [3, 4].

In the present study, the coupled Burgers’ equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which trigonometric cubic and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L₂ and L_{∞}. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

In this paper, we study the Cauchy problem for a generalized Boussinesq type equation in $\mathbb{R}^n$. We establish a dispersive estimate for the linear group associated with the generalized Boussinesq type equation. As applications, the global existence, decay and scattering of solutions are established for small initial data.

The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is analyzed. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

Using the theory of continuous Bessel wavelet transform in $L^2 (\mathbb{R})$-spaces, we established the Parseval and inversion formulas for the $L^{p,\sigma}(\mathbb{R}^+)$- spaces. We investigate continuity and boundedness properties of Bessel wavelet transform in Besov-Hankel spaces. Our main results: are the characterization of Besov-Hankel spaces by using continuous Bessel wavelet coefficient.

This paper investigates the spatial behavior of the solutions of the double-diffusive Darcy plane flow in a semi-infinite channel. Using the energy estimate method and the differential inequality technology, a differential inequality about the solutions is derived. By solving this differential inequality, it is proved that the solutions grow polynomially or decay exponentially with spatial variable. In the case of decay, we obtain the upper bound for the total energy. We also give some remarks to generalize the results of this paper.

This paper investigates the local and global existence for the inhomogeneous nonlinear Schrödinger equation with the nonlinearity λ|x|^{-b}|u|^{β}u. It is show that a global solution exists in the mass-subcritical for large data in the spaces L^{p}, p < 2 under some suitable conditions on b,β and p. The solution is established using a data-decomposition argument, two kinds of generalized Strichartz estimates in Lorentz spaces and a interpolation theorem.

Doege-Potter syndrome leads to severe and sustained symptomatic hypoglycemia and is associated with the solitary fibrous tumor. It is a rare cause and its diagnosis requires a clinical suspicion and other markers such as insulin-like growth factor-II. Here, we describe a case of a patient with intrathoracic tumor and hypoglycemia

Tuberculous (TB) infection remains one of the leading causes of death, especially in developing countries. TB liver involvement is a rare extrapulmonary manifestation of TB and is usually found as a secondary involvement to TB of the lung. Here we are presenting a case of Isolated TB liver abscess.

Non-islet cell tumor hypoglycemia (NICTH) is a rare but severe complication of malignancy. We present the case of 55 year old man who was admitted for severe hypoglycemia. The diagnosis of insulinoma was ruled out. After clinical work-ups, we made the diagnosis of metastatic HCC with production of IGF-2.

Rothmund-Thomson Syndrome is a rare autosomal recessive condition presenting usually in infancy that can be diagnosed based on time of onset, spreading and appearance of the poikiloderma.The purpose of reporting this case is to highlight the clinical approach to a child who presents with the features of poikiloderma.

COVID-19 is associated with respiratory failure and subsequent hypoxia. New therapies that reduce hypoxia are urgently needed. Molecular hydrogen (H2) therapy may reduce inflammation, oxidative stress, and hypoxia. This case report suggests that hydrogen-rich water (HRW) increases oxygen saturation increase exercise tolerance in a patient with COVID-19-like symptoms.

Endomyocardial fibrosis is a disease that causes restrictive cardiomyopathy, it varies in presentation ranging from asymptomatic to full-blown heart failure. its pathogenesis is poorly understood and it poses a diagnostic and treatment challenge due to its infrequent occurrence.

A 77-year-old woman was admitted to our emergency department complaining of abdominal pain. Computed tomography was performed and showed aerobilila and a large 5.1 cm gallstone lodged in the small intestine. She underwent emergency surgery. Intraoperative findings noted small bowel obstruction caused by a large gallstone.

Adrenergic myocarditis is a very rare manifestation of pheochromocytoma. On the other hand, due to the pandemic situation, COVID-19 pneumonia is now considered a differential diagnosis of patients with dyspnea. This case illustrates diagnostic difficulties of such uncommon disease during this pandemic situation.

Brucellosis is among the most prevalent zoonotic diseases that can involve almost any organ. Here we present a case of deep vein thrombosis due to human brucellosis. We described a 62 old male farmer with a history of brucellosis and deep vein thrombosis on his right leg

The beating of a pulmonary vein during cardiac catheterization is a rare phenomenon caused by the heart beating through the pericardial effusion when a cardiac tamponade occurs. This “beating pulmonary vein” sign is useful for early detection of a tamponade before circulatory collapse occurs.

The alignment of third molars has largely been overlooked by orthodontists. Since difficulties arise during their eruption, a plethora of surgical procedures is advocated for their management, including prophylactic extraction. This case series describes the use of midfacial development in 4 patients and illustrates successful patterns of third molar eruption.

This is a new case of simultaneous occurrences of AMN and COVID-19 in a previously healthy woman. The importance of the current report lies in increasing the awareness of all physicians about the probable association of COVID-19 with retinal vascular injury and ischemia.

We report a 54-year-old man with treatment-resistant depression (TRD) and hypothyroidism who responded to LT3/LT4 combination, rather than LT4 alone. He was able to discontinue all antidepressant medications eventually. Interestingly, the patient has a DIO2 polymorphism.

Hepatic epithelioid angiomyolipomas are uncommon mesenchymal neoplasms. They are difficult to diagnose by imaging, especially when the fatty component is scant or absent. The gold standard for the diagnosis is histologic examination coupled with an immunohistochemical study. Positive HMB-45 immunostaining of the myoid cells is a major diagnostic feature.

Mixed neuroendocrine-non-neuroendocrine neoplasms (MiNEN) are an heterogeneous subgroup of rare neoplasms. Triple componed MiNEN associating neuroendocrine tumor, adenocarcinoma and squamous cell carcinoma are rarely described in the literature. Here we present the unique case of a 56-year-old women with neuroendocrine tumor, adenocarcinoma and squamous cell carcinoma of the duodenum