Inhomogeneous parabolic equations on unbounded metric measure spaces

We study inhomogeneous semilinear parabolic equations with source term f independent of time u_{t}={\Delta}u+u^{p}+f(x) on a metric measure space, subject to the conditions that f(x)\geq 0 and u(0,x)=\phi(x)\geq 0. By establishing Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence, and global existence of weak solutions. This paper generalizes previous results on Euclidean spaces to general metric measure spaces

Coronary Microvascular Dysfunction and Microvascular Angina

Many patients present with myocardial ischemic symptoms, but fail to be diagnosed of obstructed coronary artery disease, since the normal coronary arteries or no any atherosclerosis stenosis ≥ 50% at coronary angiography. Myocardial ischemia can be caused by either abnormalities of epicardial coronary arteries or coronary microvascular dysfunction (CMD). Patient with microvascular angina in the absence of coronary artery disease and myocardial diseases, CMD is suggested to be the unique cause of symptoms. The previous clinical and pathogenetic classification of CMD is based on presence or absence of coronary artery disease, myocardial diseases, or other traditional risk factors, which would obscure the importance of the disease primarily provoked by CMD. The role of atherosclerotic plaque rupture in epicardial coronary arteries and the abnormality of hemorheology (especially in perimenopausal women) should be more stressed in the pathogenetic mechanism of CMD. The pathogenetic mechanism of CMD will be classified according to microvascular structure (embolization and stenosis), microvascular function and blood risk factors in this paper. The CMD related diseases including cardiac X syndrome and coronary slow flow would be better uniformly named as microvascular angina. While little data supported therapies for CMD related diseases so far, the blood healthy therapy as a novel method is recommended to treat microvascular angina, especially in the patients with high hematocrit, increased blood viscosity and coronary slow flow