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- Existence of weak solutions for parabolic equations involving the p(x,t)-Laplacian with non-continuous, unbounded pPublication . Bae, Hyeong-Ohk; Borges de Oliveira, Hermenegildo; Wolf, JörgWe consider a parabolic equation involving the p(x,t)-Laplacian as a prime term in a time-space cylinder QT := Ω × (0,T) ⊂ Rd ×R,d≥ 2. We assume that p(x,t): QT → (1,+∞) is a continuous function, except in a closed set of zero measure. Under the assumption that p is unbounded, such that 1 < infK×(δ,T)p(x,t) ≤ supK× (δ,T) p(x,t) < ∞ for every compact set K ⊂ Ω and δ ∈ (0,T), we prove the existence of a weak solution to the corresponding initial- and boundary-value problem for any given initial velocity in L2(Ω). © 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
- A note on varieties of ordered algebrasPublication . Clementino, Maria Manuel; Rodelo, DianaThe aim of this work is to study the notions of lax protomodular and Ord-Mal’tsev category at the level of (coherent) varieties of (pre)ordered algebras and to further compare them, as has been done in the non-ordered context. We characterise varieties of ordered algebras which are (co)lax protomodular and those which are Ord-Mal’tsev, in terms of operations of arities given by ordered sets and inequalities involving them. We exhibit examples of (co)lax protomodular non-degenerate Ordcategories, which were unknown. We prove that, for varieties of ordered algebras which are Ord-Mal’tsev categories, the order of their algebras is degenerate (i.e. is symmetric). As a consequence, the implication ``protomodular ⇒ Mal’tsev'' cannot be carried out to our context. The case of non-coherent ordered varieties which are Ord-Mal’tsev categories is also addressed, where we show the existence of algebras with non-degenerate order.