arXiv subjects

Serikbol Shaimardan

Publications and source records attributed to Serikbol Shaimardan.

12 recordsLinked to original sources

arXiv2026

Fujita exponents on quantum Euclidean spaces

We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting.

math.AP↗

arXiv2026

On the $q$-analogue of Duhamel's principle

In this paper, we revisit the classical Duhamel's principle and provide a self-contained proof of this fundamental tool for linear evolution equations and systems of coupled equations. Moreover, we establish a $q$-analogue of Duhamel's principle for $q$-evolution equations of order $k\geq 1$ generated by Jackson's $q$-difference operator.

math.AP↗

arXiv2025

Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications

In this paper, we study Hörmander type Fourier multiplier theorem and the Nikolskii inequality on quantum tori. On the way to obtain these results, we also prove some classical inequalities such as Paley type, Hausdorff-Young-Paley, Hardy-Littlewood, and Logarithmic Sobolev inequalities on quantum tori. As applications we establish embedding theorems between Sobolev, Besov spaces as well as embeddings between Besov and Wiener and Beurling spaces on quantum tori. We also analyse $β$-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces on quantum tori. As an applications of the analysis, we also derive a version of the Nash inequality, and the time decay for solutions of a heat type equation.

math.FA↗

arXiv2025

Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces

In this work, we study the Sobolev inequality on noncommutative Euclidean spaces. As a simple consequence, we obtain the Gagliardo-Nirenberg type inequality and as its application we show global well-posedness of nonlinear PDEs in the noncommutative Euclidean space. Moreover, we show that the logarithmic Sobolev inequality is equivalent to the Nash inequality for possibly different constants in this noncommutative setting by completing the list in noncommutative Varopoulos's theorem in [37]. Finally, we present a direct application of the Nash inequality to compute the time decay for solutions of the heat equation in the noncommutative setting.

math.AP↗

arXiv2023

Some estimates for Mittag-Leffler function in quantum calculus and applications

The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. In the present paper we prove the following estimate for the $q$-Mittag-Leffler function: \begin{eqnarray*} \frac{1}{1+Γ_q\left(1-α\right)z}\leq e_{α,1}\left(-z;q\right)\leq\frac{1}{1+Γ_q\left(α+1\right)^{-1}z}. \end{eqnarray*} for all $0 < α< 1$ and $z>0$. Moreover, we apply it to investigate the solvability results for direct and inverse problems for time-fractional pseudo-parabolic equations in quantum calculus for a large class of positive operators with discrete spectrum.

math.AP↗

arXiv2023

Well-posedness of heat and wave equations generated by Rubin's q-difference operator in Sobolev spaces

In this paper, we investigate difference-differential operators of parabolic and hyperbolic types. Namely, we consider non-homogenous heat and wave equations for Rubin's difference operator. Well-posedness results are obtained in appropriate Sobolev type spaces. In particular, we prove that the heat and wave equations generated by Rubin's difference operator have unique solutions. We even show that these solutions can be represented by explicit formulas

math.AP↗

arXiv2022

The Prabhakar fractional $q$-integral and $q$-differential operators, and their properties

In this paper, we have introduced the Prabhakar fractional $q$-integral and $q$-differential operators. We first study the semi-group property of the Prabhakar fractional $q$-integral operator, which allowed us to introduce the corresponding $q$-differential operator. Formulas for compositions of $q$-integral and $q$-differential operators are also presented. We show the boundedness of the Prabhakar fractional $q$-integral operator in the class of $q$-integrable functions.

math.AP↗

arXiv2022

Direct and inverse source problems for heat equation in quantum calculus

In this paper we explore the weak solutions of the Cauchy problem and an inverse source problem for the heat equation in the quantum calculus, formulated in abstract Hilbert spaces. For this we use the Fourier series expansions. Moreover, we prove the existence, uniqueness and stability of the weak solution of the inverse problem with a final determination condition. We give some examples such as the q-Sturm-Liouville problem, the q-Bessel operator, the q-deformed Hamiltonian, the fractional Sturm-Liouville operator, and the restricted fractional Laplacian, covered by our analysis.

math.AP↗

arXiv2022

Inverse problems for q-analogue of the heat equation

In this paper we explore the weak solution of a time-dependent inverse source problem and inverse initial problem for $q$-analogue of the heat equation. As an over-determination condition we have used integral type condition on space-variable (in the case of inverse source problem) and final time condition (in the case of inverse initial problem). Series form of considered problems have been obtained via the method of spectral expansions.

math.AP↗

arXiv2022

Fractional Cauchy problems associated with the bi-ordinal Hilfer fractional $q$-derivative

To study the existence and uniqueness of solutions to Cauchy-type problems for fractional q-difference equations with the bi-ordinal Hilfer fractional q-derivative which is an extension of the Hilfer fractional q-derivative. An approach is based on the equivalence of the nonlinear Cauchy-type problem with a nonlinear Volterra q-integral equation of the second kind. Applying an analog of Banach fixed point theorem we prove the uniqueness and the existence of the solution. Moreover, we present an explicit solution to the q-analog of the Cauchy problem for the linear case.

math.AP↗

arXiv2022

Wave equation for Sturm-Liouville operator with singular potentials

The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the eigenvalues and eigenfunctions of the Sturm-Liouville operator are used. Homogeneous and inhomogeneous cases of the equation are considered. Next, existence, uniqueness, and consistency theorems for a very weak solution of the wave equation with singular coefficients are proved.

math.AP↗

arXiv2020

Existence and uniqueness of some Cauchy Type Problems in fractional q-difference calculus

In this paper we derive a sufficient condition for the existence of a unique solution of a Cauchy type q-fractional problem (involving the fractional q-derivative of Riemann-Liouville type) for some nonlinear differential equations. The key technique is to first prove that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. Moreover, we define the $q$-analogue of the Hilfer fractional derivative or composite fractional derivative operator and prove some similar new equivalence, existence and uniqueness results as above. Finally, some examples are presented to illustrate our main results in cases where we can even give concrete formulas for these unique solutions.

math.AP↗