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Bolys Sabitbek

Publications and source records attributed to Bolys Sabitbek.

15 recordsLinked to original sources
arXiv2026

Asymptotics and Scattering for Critically Weakly Hyperbolic and Singular Systems

We study a very general class of first-order linear hyperbolic systems that both become weakly hyperbolic and contain lower-order coefficients that blow up at a single time $t = 0$. In "critical" weakly hyperbolic settings, it is well-known that solutions lose a finite amount of regularity at the degenerate time $t = 0$. In this paper, we both improve upon the results in the weakly hyperbolic setting, and we extend this analysis to systems containing critically singular coefficients, which may also exhibit significantly modified asymptotics at $t = 0$. In particular, we give precise quantifications for (1) the asymptotics of solutions as $t$ approaches $0$; (2) the scattering problem of solving the system with asymptotic data at $t = 0$; and (3) the loss of regularity due to the degeneracies at $t = 0$. Finally, we discuss a variety of applications for these results, including to weakly hyperbolic and singular wave equations, equations of higher order, and equations arising from relativity and cosmology, e.g. at big bang singularities.

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arXiv2024

$C^\infty$ well-posedness of higher order hyperbolic pseudo-differential equations with multiplicities

In this paper, we study higher order hyperbolic pseudo-differential equations with variable multiplicities. We work in arbitrary space dimension and we assume that the principal part is time-dependent only. We identify sufficient conditions on the roots and the lower order terms (Levi conditions) under which the corresponding Cauchy problem is $C^\infty$ well-posed. This is achieved via transformation into a first order system, reduction into upper-triangular form and application of suitable Fourier integral operator methods previously developed for hyperbolic non-diagonalisable systems. We also discuss how our result compares with the literature on second and third order hyperbolic equations.

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arXiv2024

Global existence and blow-up of solutions to the double nonlinear porous medium equation

In this study, we examine a double nonlinear porous medium equation subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial energy. By introducing a set of potential wells, we construct invariant sets of solutions for the double nonlinear porous medium equation. For subcritical and critical initial energy scenarios, we derive the global existence and asymptotic behavior of weak solutions, as well as blow-up phenomena occurring within a finite time for the positive solution to the double nonlinear porous medium equation.

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arXiv2023

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, III: singular coefficients

In this paper we continue the analysis of non-diagonalisable hyperbolic systems initiated in \cite{GarJRuz, GarJRuz2}. Here we assume that the system has discontinuous coefficients or more in general distributional coefficients. Well-posedness is proven in the very weak sense for systems with singularities with respect to the space variable or the time variable. Consistency with the classical theory is proven in the case of smooth coefficients.

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arXiv2022

Blow-up solutions of damped Klein-Gordon equation on the Heisenberg group

Inthisnote,weprovetheblow-upofsolutionsofthesemilineardamped Klein-Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper [21] by the first author and Tokmagambetov, where the global in time well-posedness was proved for the small energy solutions.

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arXiv2021

Global existence and nonexistence of semilinear wave equation with a new condition

In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition $$α\int_0^{u } f(s)ds \leq uf(u) + βu^2 +ασ,$$ for some positive constants $α$, $β$, and $σ$, where $β< \frac{λ_1(α-2)}{2}$ with $λ_1$ being a first eigenvalue of Laplacian. By introducing a family of potential wells, we establish the invariant sets, vacuum isolation of solutions, global existence and blow-up solutions of semilinear wave equation for initial conditions $E(0)<d$ and $E(0)=d$.

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arXiv2021

Hardy and Rellich inequalities with Bessel pairs

In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $Ω\subset \mathbb{R}^n$ in order to have the following Hardy inequality \begin{equation*} \int_Ω W(x) |\nabla u|_A^2 dx \geq \int_Ω |\nabla d|^2_AH(x)|u|^2 dx, \,\,\, u \in C^{1}_0(Ω), \end{equation*} where $d(x)$ is a suitable quasi-norm (gauge), $|ξ|^2_A = \langle A(x)ξ, ξ\rangle$ for $ξ\in \mathbb{R}^n$ and $A(x)$ is an $n\times n$ symmetric, uniformly positive definite matrix defined on a bounded domain $Ω\subset \mathbb{R}^n$. We also give its $L^p$ analogue. As a consequence, we present examples for a standard Laplacian on $\mathbb{R}^n$, Baouendi-Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions $(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam \cite{GM_book}.

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arXiv2020

Principal frequency of p-sub-Laplacians for general vector fields

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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arXiv2020

Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub- Laplacians which are operators of the form $$ \mathcal{L}_p f := \sum_{i=1}^{N}X_i(|X_if|^{p_i-2}X_i f), \quad 1<p_i< \infty, $$ where $X_i, i=1,\ldots, N,$ are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups.

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arXiv2019

Geometric Hardy inequalities on starshaped sets

In this paper, we present the geometric Hardy inequalities on the starshaped sets in the Carnot groups. Also, we obtain the geometric Hardy inequalities on half-spaces for general vector fields.

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arXiv2018

Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups

In this paper, we present the geometric Hardy inequality for the sub-Laplacian in the half-spaces on the stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space on the Heisenberg group with a sharp constant \begin{equation*} \int_{\mathbb{H}^+} |\nabla_{H}u|^p dξ\geq \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(ξ)^p}{dist(ξ,\partial \mathbb{H}^+)^p} |u|^p dξ, \,\, p>1, \end{equation*} which solves the conjecture in the paper \cite{Larson}. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space on the Heisenberg group \begin{equation*} \left(\int_{\mathbb{H}^+} |\nabla_{H} u|^p dξ- \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(ξ)^p}{dist(ξ,\partial \mathbb{H}^+)^p} |u|^p dξ\right)^{\frac{1}{p}} \geq C \left(\int_{\mathbb{H}^+} |u|^{p^*} dξ\right)^{\frac{1}{p^*}}, \end{equation*} where $dist(ξ,\partial \mathbb{H}^+)$ is the Euclidean distance to the boundary, $p^* := Qp/(Q-p)$, $2\leq p<Q$, and $$\mathcal{W}(ξ)=\left(\sum_{i=1}^{n}\langle X_i(ξ), ν\rangle^2+\langle Y_i(ξ), ν\rangle^2\right)^{\frac{1}{2}},$$ is the angle function. For $p=2$, this gives the Hardy-Sobolev-Maz'ya inequality on the Heisenberg group.

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arXiv2017

Weighted $L^p$-Hardy and $L^p$-Rellich inequalities with boundary terms on stratified Lie groups

In this paper, generalised weighted $L^p$-Hardy,$ L^p$-Caffarelli-Kohn-Nirenberg, and $L^p$-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg- Pauli-Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted $L^2$-Rellich type inequality with the boundary term is obtained.

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