We study a continuation criterion for strong solutions to a magnetovisco elastic system. This system couples the Navier-Stokes equations with evolution equations for the deformation gradient and the magnetization arising from a special case of the micromagnetic energy.
Based on the multilinear Dunkl–Calderón–Zygmund operator and the classical multilinear strongly singular integral operator, this article introduces the multilinear strongly singular Dunkl–Calderón–Zygmund operator, which exhibits stronger singularity near the diagonal compared to the classical Dunkl–Calderón–Zygmund analogue. Furthermore, a new size condition is proposed, which is a localized estimate associated with the Dunkl root system $R$ and is independent of the condition for the classical multilinear Dunkl–Calderón–Zygmund operator. As the main result, the boundedness on weighted Lebesgue spaces for the multilinear strongly singular Dunkl–Calderón–Zygmund operator is established, along with its multilinear commutators and iterated commutators.
In this paper, by introducing the harmonic complex Lichtenberg sequence, we extend the classical complex Lichtenberg sequence through the incorporation of the harmonic number properties. The generating function plays a central role in establishing the analytical structure of the sequence, enabling the derivation of the closed-form expressions. This new sequence is rigorously defined using its generating function, from which we derive a determinantal formula for the complex Lichtenberg number. Furthermore, by applying the discrete difference operator, we develop a summation identity that characterizes the behavior of the harmonic complex Lichtenberg numbers, offering insight into their recursive and additive properties. In addition, we explore the matrix implications of the harmonic complex Lichtenberg sequence by constructing a special max matrix, whose elements are composed of these numbers. For this matrix, we derive explicit expressions for both its determinant and permanent, which provide critical information about the matrix's structural and its algebraic properties. These results contribute to a deeper understanding of the interplay between the number sequences and matrix theory. To support the theoretical findings, we include two detailed numerical examples that illustrate the practical computation of the determinant and permanent, thereby validating the correctness and applicability of our proposed formulas.
The stability condition in the sense of Kato is extended to a family of operators that are not densely defined even in a reflexive Banach space. A generation theorem of evolution operators is established under a new type of stability condition, so that it not only extends the well-posedness results for the inhomogeneous equation $u'(t) = Au(t) + f(t)$ with $f \in L^{p}(0,T;X)$ by the theory of integrated semigroups, but also can be applied to size-structured population models in $L^{p}$. The well-posedness for inhomogeneous equations is also characterized in terms of resolvents.
This paper studies subcritical and critical fractional $p$-Laplacian problems with vanishing potentials at infinity \[ (-\Delta)^{s}_{p}u + V(x) |u|^{p-2}u = M(x) f(u) + K(x) |u|^{p^{*}_{s}-2}u, \quad x \in \mathbb{R}^{N}, \] where $s \in (0,1)$, $p \in (1,+\infty)$, $N \gt sp$, $p^{*}_{s} = Np/(N-sp)$ denotes the fractional critical exponent, and $(-\Delta)^{s}_{p}$ stands for the fractional $p$-Laplacian operator. Under some appropriate conditions for $V$, $M$, $K$ and $f$, we obtain the existence results of positive solutions by using variational methods.
