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The paper is concerned with gradient blowup behavior for a semilinear parabolic equation u t − Δ u = δ m (x) | ∇ u | p + Λ in Ω × (0 , T) with the zero Dirichlet condition. Two results with p > m + 2 and m ⩾ 0 are established. One of which is that any gradient blowup solution follows a global ODE type behavior, with domination of normal derivatives over the tangential derivatives. The other is about time-increasing solutions. Zhang and Hu (2010) [50] obtained the precise gradient blowup rate in one-dimensional case, but the higher dimensional case was left as an open problem. Here we solve this problem by establishing the gradient blowup rate, for any small γ > 0 , C (T − t) − m + 1 p − m − 2 ⩽ ‖ ∇ u ‖ ∞ ⩽ C γ (T − t) − m + 1 p − m − 2 − γ for suitable ranges of p and m , which extends the result of (Attouchi and Souplet (2020) [3]) to the case m ≠ 0. As an important by-product which is of independent interests itself, the gradient estimate near boundary for the corresponding elliptic equation is derived under weaker assumptions on the inhomogeneous term. [ABSTRACT FROM AUTHOR]

Background: Prenatal diagnosis of Berry syndrome, a rare combination of cardiac anomalies including aortopulmonary window (APW), aortic origin of the right pulmonary artery (RPA), interrupted aortic arch (IAA), hypoplastic aortic arch, or coarctation of the aorta (COA), poses a significant challenge. Due to the rarity of the disease, and the limited case reports available to features the complex malformation of Berry syndrome postpartum, this article introduces an innovative approach to visually showcase this unusual disease. The proposed method provides a comprehensive display of the structural deformities, offering valuable insights for clinical practitioners seeking to comprehend this condition. Case presentation: In this report, we present a case where fetal echocardiography aided in diagnosing Berry syndrome, which was later confirmed through postpartum cardiovascular casting. Our experience highlights the importance of using the three-vessel view to diagnose APW and aortic origin of the right pulmonary artery. Additionally, obtaining true cross-sectional and sagittal views by continuously scanning from the three-vessel-trachea view to the long-axis view of the aortic arch is necessary to image IAA or coarctation of the aortic arch. Conclusions: Early and accurate prenatal diagnosis of Berry syndrome is feasible and our cardiovascular cast can perfectly display the microvascular morphology of the fetal heart, which may have great application prospects for postpartum diagnosis and teaching of complex cardiac abnormalities. [ABSTRACT FROM AUTHOR]

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the m -Laplacian operator − Δ m u = u q | ∇ u | p in R N , where N ⩾ 1 , m > 1 and p , q ⩾ 0. The technique of Bernstein gradient estimates is utilized to study the case p < m. Moreover, a Liouville-type theorem for supersolutions under subcritical range of exponents q (N − m) + p (N − 1) < N (m − 1) is also established. Then, we use a degree argument to obtain the existence of positive weak solutions for a nonlinear Dirichlet problem of the type − Δ m u = f (x , u , ∇ u) , with f satisfying certain structure conditions. Our proof is based on a priori estimates, which will be accomplished by using a blow-up argument together with the Liouville-type theorem in the half-space. As another application, some new Harnack inequalities are proved. [ABSTRACT FROM AUTHOR]