Well-posed results for nonlocal biparabolic equation with linear and nonlinear source terms

In this paper, we consider the biparabolic problem under nonlocal conditions with both linear and nonlinear source terms. We derive the regularity property of the mild solution for the linear source term while we apply the Banach fixed-point theorem to study the existence and uniqueness of the mild solution for the nonlinear source term. In both cases, we show that the mild solution of our problem converges to the solution of an initial value problem as the parameter epsilon tends to zero. The novelty in our study can be considered as one of the first results on biparabolic equations with nonlocal conditions.

The number of representations of squares by integral quaternary quadratic forms

Let [Formula: see text] be a positive definite (non-classic) integral quaternary quadratic form. We say [Formula: see text] is strongly[Formula: see text]-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this paper, we show that there are exactly [Formula: see text] strongly [Formula: see text]-regular diagonal quaternary quadratic forms representing [Formula: see text] (see Table [Formula: see text]). In particular, we use eta-quotients to prove the strong [Formula: see text]-regularity of the quaternary quadratic form [Formula: see text], which is, in fact, of class number [Formula: see text] (see Lemma 4.5 and Proposition 4.6).

Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

In this paper, we consider a class of optimal control problems governed by a differential system. We analyse the sensitivity relations satisfied by the co-state arc of the Pontryagin maximum principle and the value function that associates the optimal value of the control problem to the initial time and state. Such a relationship has been already investigated for state-constrained problems under some controllability assumptions to guarantee Lipschitz regularity property of the value function. Here, we consider the case with intermediate and final state constraints, without any controllability assumption on the system, and without Lipschitz regularity of the value function. Because of this lack of regularity, the sensitivity relations cannot be expressed with the sub-differentials of the value function. This work shows that the constrained problem can be reformulated with an auxiliary value function which is more regular and suitable to express the sensitivity of the adjoint arc of the original state-constrained control problem along an optimal trajectory. Furthermore, our analysis covers the case of normal optimal solutions, and abnormal solutions as well.

Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

We adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.

R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay

In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then we apply these estimates to stochastic convolutions incurred by distributed delay to study their regularity property. Last, we present some easily-verified results by considering the regularity of a class of systems whose delay operators have the same order derivatives as those in instantaneous ones.

GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

On a repulsion Keller–Segel system with a logarithmic sensitivity

In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T > 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.

On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space

<p style='text-indent:20px;'>We consider in this paper the nonlinear elliptic equation with Neumann boundary condition</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \begin{cases} \Delta u = a|u|^{m-1}u\, \, \mbox{ in }\, \, \mathbb{R}^{n+1}_{+}\\ \dfrac{\partial u}{\partial t} = b|u|^{\eta-1}u+f\, \, \mbox{ on }\, \, \partial \mathbb{R}^{n+1}_{+}. \end{cases} \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>For <inline-formula><tex-math id="M1">\begin{document}$ a, b\neq 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ m&gt;\frac{n+1}{n-1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ (n&gt;1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \eta = \frac{m+1}{2} $\end{document}</tex-math></inline-formula> and small data <inline-formula><tex-math id="M5">\begin{document}$ f\in L^{\frac{nq}{n+1}, \infty}(\partial \mathbb{R}^{n+1}_{+}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ q = \frac{(n+1)(m-1)}{m+1} $\end{document}</tex-math></inline-formula> we prove that the problem is solvable. More precisely, we establish existence, uniqueness and continuous dependence of solutions on the boundary data <inline-formula><tex-math id="M7">\begin{document}$ f $\end{document}</tex-math></inline-formula> in the function space <inline-formula><tex-math id="M8">\begin{document}$ \mathbf{X}^{q}_{\infty} $\end{document}</tex-math></inline-formula> where</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \|u\|_{ \mathbf{X}^{q}_{\infty}} = \sup\limits_{t&gt;0}t^{\frac{n+1}{q}-1}\|u(t)\|_{L^{\infty}( \mathbb{R}^{n})}+\|u\|_{L^{\frac{q(m+1)}{2}, \infty}( \mathbb{R}^{n+1}_{+})}+\|\nabla u\|_{L^{q, \infty}( \mathbb{R}^{n+1}_{+})}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>As a direct consequence, we obtain the local regularity property <inline-formula><tex-math id="M9">\begin{document}$ C^{1, \nu}_{loc} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \nu\in (0, 1) $\end{document}</tex-math></inline-formula> of these solutions as well as energy estimates for certain values of <inline-formula><tex-math id="M11">\begin{document}$ m $\end{document}</tex-math></inline-formula>. Boundary values decaying faster than <inline-formula><tex-math id="M12">\begin{document}$ |x|^{-(m+1)/(m-1)} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ x\in \mathbb{R}^{n}\setminus\{0\} $\end{document}</tex-math></inline-formula> yield solvability and this decay property is shown to be sharp for positive nonlinearities.</p><p style='text-indent:20px;'>Moreover, we are able to show that solutions inherit qualitative features of the boundary data such as positivity, rotational symmetry with respect to the <inline-formula><tex-math id="M14">\begin{document}$ (n+1) $\end{document}</tex-math></inline-formula>-axis, radial monotonicity in the tangential variable and homogeneity. When <inline-formula><tex-math id="M15">\begin{document}$ a, b&gt;0 $\end{document}</tex-math></inline-formula>, the critical exponent <inline-formula><tex-math id="M16">\begin{document}$ m_c $\end{document}</tex-math></inline-formula> for the existence of positive solutions is identified, <inline-formula><tex-math id="M17">\begin{document}$ m_c = (n+1)/(n-1) $\end{document}</tex-math></inline-formula>.</p>