Effects of reactive gradient term in a multi-nonlinear parabolic problem

AbstractThis paper deals with parabolic equation ut=Δu+|∇u|r−aepu subject to nonlinear boundary flux ∂u/∂η=equ, where r>1, p,q,a>0. There are two positive sources (the gradient reaction and the boundary flux) and a negative one (the absorption) in the model. It is well known that blow-up or not of solutions depends on which one dominating the model, the positive or negative sources, and furthermore on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of the reactive gradient term on the asymptotic behavior of solutions. We at first determine the critical blow-up exponent, and then obtain the blow-up rate, the blow-up set as well as the spatial blow-up profile for blow-up solutions in the one-dimensional case. It turns out that the gradient term makes a substantial contribution to the formation of blow-up if and only if r⩾2, where the critical r=2 is such a balance situation of the two positive sources for which the effects of the gradient reaction and the boundary source are at the same level. In addition, it is observed that the gradient term with r>2 significantly affects the blow-up rate also. In fact, the gained blow-up rates themselves contain the exponent r of the gradient term. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. As for the influence of gradient perturbations on spatial blow-up profiles, we only need some coefficients related to r for the profile estimates, while the exponent of the profile itself is r-independent. This seems natural for boundary blow-up solutions that the spatial profiles mainly rely on the exponent of the boundary singularity

Conformal perturbations of dirac operators and general Kastler-Kalau-Walze type theorems for even dimensional manifolds with boundary

In this paper, we establish the proof of general Kastler-Kalau-Walze type theorems for conformal perturbations of dirac Operators on even dimensional compact manifolds with (respectively without) boundary.Comment: arXiv admin note: substantial text overlap with arXiv:2310.09775; text overlap with arXiv:2111.1503

Inactivating hepatic follistatin alleviates hyperglycemia

Unsuppressed hepatic glucose production (HGP) contributes substantially to glucose intolerance and diabetes, which can be modeled by the genetic inactivation of hepatic insulin receptor substrate 1 (Irs1) and Irs2 (LDKO mice). We previously showed that glucose intolerance in LDKO mice is resolved by hepatic inactivation of the transcription factor FoxO1 (that is, LTKO mice)-even though the liver remains insensitive to insulin. Here, we report that insulin sensitivity in the white adipose tissue of LDKO mice is also impaired but is restored in LTKO mice in conjunction with normal suppression of HGP by insulin. To establish the mechanism by which white adipose tissue insulin signaling and HGP was regulated by hepatic FoxO1, we identified putative hepatokines-including excess follistatin (Fst)-that were dysregulated in LDKO mice but normalized in LTKO mice. Knockdown of hepatic Fst in the LDKO mouse liver restored glucose tolerance, white adipose tissue insulin signaling and the suppression of HGP by insulin; however, the expression of Fst in the liver of healthy LTKO mice had the opposite effect. Of potential clinical significance, knockdown of Fst also improved glucose tolerance in high-fat-fed obese mice, and the level of serum Fst was reduced in parallel with glycated hemoglobin in obese individuals with diabetes who underwent therapeutic gastric bypass surgery. We conclude that Fst is a pathological hepatokine that might be targeted for diabetes therapy during hepatic insulin resistance

Modified Novikov Operators and the Kastler-Kalau-Walze-Type Theorem for Manifolds with Boundary

In this paper, we give two Lichnerowicz-type formulas for modified Novikov operators. We prove Kastler-Kalau-Walze-type theorems for modified Novikov operators on compact manifolds with (respectively without) a boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds