## Publications

330 results found

Chatzakou M, Ruzhansky M, Tokmagambetov N, 2022, The Heat Equation with Singular Potentials. II: Hypoelliptic Case, *ACTA APPLICANDAE MATHEMATICAE*, Vol: 179, ISSN: 0167-8019

Ruzhansky M, Sabitbek B, Torebek B, 2022, Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified groups, *manuscripta mathematica*, ISSN: 0025-2611

<jats:title>Abstract</jats:title><jats:p>In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in Ruzhansky and Suragan (J Differ Eq 262:1799–1821, 2017) for stratified groups.</jats:p>

Dasgupta A, Ruzhansky M, Tushir A, 2022, Discrete time-dependent wave equations I. Semiclassical analysis, *Journal of Differential Equations*, Vol: 317, Pages: 89-120, ISSN: 0022-0396

In this paper we consider a semiclassical version of the wave equations with singular Hölder time-dependent propagation speeds on the lattice ħZn. We allow the propagation speed to vanish leading to the weakly hyperbolic nature of the equations. Curiously, very much contrary to the Euclidean case considered by Colombini, de Giorgi and Spagnolo [2] and by other authors, the Cauchy problem in this case is well-posed in ℓ2(ħZn). However, we also recover the well-posedness results in the intersection of certain Gevrey and Sobolev spaces in the limit of the semiclassical parameter ħ→0.

Ruzhansky M, Yessirkegenov N, 2022, Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature, *Journal of Mathematical Analysis and Applications*, Vol: 507, ISSN: 0022-247X

In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces. Moreover, in some cases we also show their equivalence with Trudinger-Moser inequalities. As consequences, the relations between the constants of these inequalities are investigated yielding asymptotically best constants in the obtained inequalities. We also obtain the corresponding uncertainty type principles.

Ruzhansky M, Yessirkegenov N, 2022, A comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups, *Nonlinear Analysis, Theory, Methods and Applications*, Vol: 215, ISSN: 0362-546X

In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in t-boundedness of solutions of nonlinear equations for the heat p-sub-Laplacian on stratified Lie groups.

Chatzakou M, Ruzhansky M, Tokmagambetov N, 2022, Fractional SchrÖdinger Equations with Singular Potentials of Higher Order. II: Hypoelliptic Case, *Reports on Mathematical Physics*, Vol: 89, Pages: 59-79, ISSN: 0034-4877

In this paper we consider the space-fractional Schrödinger equation with a singular potential for a wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in the setting of graded Lie groups. The paper is a continuation and extension of the first part [2] where the classical Schrödinger equation on Rn with singular potentials was considered.

Ruzhansky M, Yessirkegenov N, 2022, Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups, *Journal of Differential Equations*, Vol: 308, Pages: 455-473, ISSN: 0022-0396

In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold M: {ut−LMu=f(u),x∈M,t>0,u(0,x)=u0(x),x∈M, for u0≥0, where LM is a sub-Laplacian of M. In the case when M is a connected unimodular Lie group G, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with u0≢0, blow up in finite time if and only if 1<p≤pF:=1+2/D when f(u)≃up, where D is the global dimension of G. In the case 1<p<pF and when f:[0,∞)→[0,∞) is a locally integrable function such that f(u)≥K2up for some K2>0, we also show that the differential inequality ut−LMu≥f(u) does not admit any nontrivial distributional (a function u∈Llocp(Q) which satisfies the differential inequality in D′(Q)) solution u≥0 in Q:=(0,∞)×G. Furthermore, in the case when G has exponential volume growth and f:[0,∞)→[0,∞) is a continuous increasing function such that f(u)≤K1up for some K1>0, we prove that the Cauchy problem has a global, classical solution for 1<p<∞ and some positive u0∈Lq(G) with 1≤q<∞. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds M.

Karimov E, Ruzhansky M, Tokmagambetov N, 2022, Cauchy type problems for fractional differential equations, *Integral Transforms and Special Functions*, Vol: 33, Pages: 47-64, ISSN: 1065-2469

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann–Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann–Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type of initial, inner, and inner-boundary conditions is also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.

Chatzakou M, Ruzhansky M, Tokmagambetov N, 2022, Fractional Klein-Gordon equation with singular mass. II: hypoelliptic case, *Complex Variables and Elliptic Equations*, Vol: 67, Pages: 615-632, ISSN: 1747-6933

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part [Altybay et al. Fractional Klein-Gordon equation with singular mass. Chaos Solitons Fractals. 2021;143:Article ID 110579] which was devoted to the classical Euclidean Klein-Gordon equation.

Bin-Saad MG, Hasanov A, Ruzhansky M, 2022, Some properties relating to the Mittag–Leffler function of two variables, *Integral Transforms and Special Functions*, Vol: 33, Pages: 400-418, ISSN: 1065-2469

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integral representation, and its relationship with Wright hypergeometric function are investigated and established. Also, properties of the Mittag–Leffler function of two variables associated with fractional calculus operators are considered.

Avetisyan Z, Grigoryan M, Ruzhansky M, 2021, Approximations in L<sup>1</sup> with convergent Fourier series, *Mathematische Zeitschrift*, Vol: 299, Pages: 1907-1927, ISSN: 0025-5874

For a separable finite diffuse measure space M and an orthonormal basis { φn} of L2(M) consisting of bounded functions φn∈ L∞(M) , we find a measurable subset E⊂ M of arbitrarily small complement | M\ E| < ϵ, such that every measurable function f∈ L1(M) has an approximant g∈ L1(M) with g= f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of M= G/ H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

Cardona D, Delgado J, Ruzhansky M, 2021, L<sup>p</sup> -Bounds for Pseudo-differential Operators on Graded Lie Groups, *Journal of Geometric Analysis*, Vol: 31, Pages: 11603-11647, ISSN: 1050-6926

In this work we obtain sharp Lp-estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to Hörmander on Rn. The main result extends the classical Fefferman’s sharp theorem on the Lp-boundedness of pseudo-differential operators for Hörmander classes on Rn to general graded Lie groups, also adding the borderline ρ= δ case.

Kumar V, Ruzhansky M, 2021, <i>L</i> <i>p</i>-<i>L</i> <i>q</i> Boundedness of <i>(k, a)</i>-Fourier Multipliers with Applications to Nonlinear Equations, *International Mathematics Research Notices*, ISSN: 1073-7928

<jats:title>Abstract</jats:title> <jats:p>The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator. The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness, we first establish Paley inequality and Hausdorff–Young–Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations.</jats:p>

Ruzhansky M, Torebek BT, 2021, Van der Corput lemmas for Mittag-Leffler functions. II. α–directions, *Bulletin des Sciences Mathematiques*, Vol: 171, ISSN: 0007-4497

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form Iα,β(λ)=∫REα,β(iαλϕ(x))ψ(x)dx, for the range 0<α≤2,β>0. This extends the variety of estimates obtained in the first part, where integrals with functions Eα,β(iλϕ(x)) have been studied. Several generalisations of the van der Corput lemmas are proved. As an application of the above results, the generalised Riemann-Lebesgue lemma, the Cauchy problem for the time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

Delgado J, Ruzhansky M, 2021, Schatten-von Neumann classes of integral operators, *Journal des Mathematiques Pures et Appliquees*, Vol: 154, Pages: 1-29, ISSN: 0021-7824

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in Rn of finite measure, and conditions for operators on Rn given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.

Restrepo JE, Ruzhansky M, Suragan D, 2021, Explicit solutions for linear variable–coefficient fractional differential equations with respect to functions, *Applied Mathematics and Computation*, Vol: 403, ISSN: 0096-3003

Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula [1, Theorem 4.1] to a general class of linear fractional differential equations with variable coefficients, to complex fractional derivatives, and to fractional derivatives with respect to a given function.

Chatzakou M, Delgado J, Ruzhansky M, 2021, On aclass of anharmonic oscillators, *JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES*, Vol: 153, Pages: 1-29, ISSN: 0021-7824

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- Citations: 4

Ruzhansky M, Velasquez-Rodriguez JP, 2021, Non-harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary, *MATHEMATISCHE NACHRICHTEN*, Vol: 294, Pages: 1783-1820, ISSN: 0025-584X

Dasgupta A, Ruzhansky M, 2021, Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality (vol 27, 15, 2021), *JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS*, Vol: 27, ISSN: 1069-5869

Ruzhansky M, Verma D, 2021, Hardy inequalities on metric measure spaces, II: the case p > q, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 477, ISSN: 1364-5021

Altybay A, Ruzhansky M, Sebih ME,
et al., 2021, The heat equation with strongly singular potentials, *Applied Mathematics and Computation*, Vol: 399, Pages: 1-15, ISSN: 0096-3003

In this paper we consider the heat equation with strongly singular potentials and prove that it has a ”very weak solution”. Moreover, we show the uniqueness and consistency results in some appropriate sense. The cases of positive and negative potentials are studied. Numerical simulations are done: one suggests so-called ”laser heating and cooling” effects depending on a sign of the potential. The latter is justified by the physical observations.

Kumar V, Ruzhansky M, 2021, A note on K-functional, Modulus of smoothness, Jackson theorem and Bernstein–Nikolskii–Stechkin inequality on Damek–Ricci spaces, *Journal of Approximation Theory*, Vol: 264, ISSN: 0021-9045

In this paper we study approximation theorems for L2-space on Damek–Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek–Ricci spaces. We also prove Bernstein–Nikolskii–Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we prove equivalence of the K-functional and modulus of smoothness for Damek–Ricci spaces.

Dasgupta A, Ruzhansky M, 2021, Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality, *Journal of Fourier Analysis and Applications*, Vol: 27, ISSN: 1069-5869

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our papers (Dasgupta and Ruzhansky in Trans Am Math Soc 368(12):8481–8498, 2016) and (Dasgupta and Ruzhansky in Trans Am Math Soc Ser B 5:81–101, 2018). We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.

Ruzhansky M, Taranto CA, 2021, Time-Dependent Wave Equations on Graded Groups, *ACTA APPLICANDAE MATHEMATICAE*, Vol: 171, ISSN: 0167-8019

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- Citations: 4

Altybay A, Ruzhansky M, Sebih ME,
et al., 2021, Fractional Schrödinger Equation with Singular Potentials of Higher Order, *Reports on Mathematical Physics*, Vol: 87, Pages: 129-144, ISSN: 0034-4877

In this paper the space-fractional Schrödinger equations with singular potentials are studied. Delta like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of ‘weakened’ solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particles accumulating effect is observed in the singular cases. From the mathematical point of view a “splitting of the strong singularity” phenomena is also observed.

Altybay A, Ruzhansky M, Sebih ME,
et al., 2021, Fractional Klein-Gordon equation with singular mass, *Chaos, Solitons and Fractals*, Vol: 143, ISSN: 0960-0779

We consider a space-fractional wave equation with a singular mass term depending on the position and prove that it is very weak well-posed. The uniqueness is proved in some appropriate sense. Moreover, we prove the consistency of the very weak solution with classical solutions when they exist. In order to study the behaviour of the very weak solution near the singularities of the coefficient, some numerical experiments are conducted where the appearance of a wall effect for the singular masses of the strength of δ2 is observed.

Kirilov A, de Moraes WAA, Ruzhansky M, 2021, Global hypoellipticity and global solvability for vector fields on compact Lie groups, *JOURNAL OF FUNCTIONAL ANALYSIS*, Vol: 280, ISSN: 0022-1236

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- Citations: 2

Ashyralyev A, Kalmenov TS, Ruzhansky MV, et al., 2021, Preface, Pages: v-vi, ISSN: 2194-1009

Fischer V, Ruzhansky M, 2021, Fourier multipliers on graded lie groups, *Colloquium Mathematicum*, Vol: 165, Pages: 1-30, ISSN: 0010-1354

We study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that Hörmander-type conditions on the Fourier multipliers imply Lp-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper.

Ruzhansky M, Sabitbek B, Suragan D, 2021, Principal frequency of p-sub-Laplacians for general vector fields, *Zeitschrift fur Analysis und ihre Anwendung*, Vol: 40, Pages: 97-109, ISSN: 0232-2064

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

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