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In this note we try to understand the blow-up of solutions to Nakao's problem by using nonlinear ordinary differential inequalities.

This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about…

We analyze a system of linear algebraic equations whose solutions lead to a proof of a generalization of Boole's formula. In particular, our approach provides an elementary and short alternative to Katsuura's proof of this generalization.

This paper is devoted to strictly hyperbolic systems and equations with non-smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of…

In this paper, we bring a complete solution to the Ovals problem, as formulated in [3] and [24].

In this note we provide a new and efficient approach to uniform estimates for solutions to complex Monge-Ampere equations, as well as for solutions to geometric PDE's that satisfy a determinantal majorization.

A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…

In this paper, we study the form type Calabi-Yau equation. We define the astheno-Ricci curvature and prove that there exists a solution for the form type Calabi-Yau equation if the astheno-Ricci curvature is non-positive.

We construct a universal trigonometric solution of the Gervais-Neveu-Felder equation in the case of finite dimensional simple Lie algebras and finite dimensional contragredient simple Lie superalgebras.

In this letter we present constant solutions to the tetrahedron equations proposed by Zamolodchikov. In general, from a given solution of the Yang-Baxter equation there are two ways to construct solutions to the tetrahedron equation. There…

We give two applications of the Aleksandrov-Bakelman-Pucci estimate to the Calabi-Yau equation on symplectic four-manifolds. The first is solvability of the equation on the Kodaira-Thurston manifold for certain almost-Kahler structures…

Various methods to find Calabi-Yau differential equations are discussed.

We consider the natural generalization of the parabolic Monge-Amp\`ere equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex manifold is locally flat and admits a…

Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5).

By adopting the 5D version of the Wu-Yang Ansatz we present in closed form a black hole solution in the Einstein-Yang-Mills-Gauss-Bonnet (EYMGB) theory. In the EYM limit, we recover the 5D black hole solution already known.

In this paper we introduce a new equation on the compact Kahler manifolds. Solution of this equation corresponds to the Calabi-Yau metric. New equation differs from the Monge--Ampere equation considered by Calabi and Yau.

We obtain a generalized Euler-Lagrange differential equation and transversality optimality conditions for Herglotz-type higher-order variational problems. Illustrative examples of the new results are given.

In this paper, we consider the Yamabe equation on a complete noncompact Riemannian manifold and find some geometric conditions on the manifold such that the Yamabe problem admits a bounded positive solution.

In this survey we consider generalizations for Young and Cauchy--Bunyakowsky inequalities with applications.

We prove that any asymptotically flat solution to the spherically symmetric SU(2) Einstein-Yang/Mills equations is globally defined. This result applies in particular to the interior of colored black holes.