arXiv subjects

Bessem Samet

Publications and source records attributed to Bessem Samet.

At least 19 recordsLinked to original sources

arXiv2026

A capacitary approach to Lyapunov-type inequalities for elliptic problems on weighted graphs

We initiate the study of Lyapunov-type inequalities for Dirichlet problems driven by the discrete p-Laplacian on weighted graphs. The approach is capacitary and is based on point p-capacities and the associated capacitary radii. First, we prove general Lyapunov-type inequalities on arbitrary connected locally finite weighted graphs. These inequalities provide intrinsic lower bounds, expressed in terms of the capacitary radii, for the positive part of the potential whenever the corresponding Dirichlet problem admits a nontrivial solution. Next, we estimate these capacitary radii in several geometric settings and prove the sharpness of the resulting Lyapunov-type inequalities. As an application, we derive lower bounds for the first weighted Dirichlet eigenvalue of the discrete p-Laplacian.

math.AP↗

arXiv2026

A Capacitary Approach to Semilinear Elliptic Inequalities with Potentials on Weighted Graphs

We develop a capacitary approach to semilinear elliptic inequalities on weighted graphs with a potential. More precisely, we study the nonexistence of nontrivial nonnegative solutions of \[ \Delta u+w(x)u+v(x)u^\sigma\le0 \qquad\text{in }V, \] where $(V,\omega,\mu)$ is a connected, locally finite weighted graph, $\Delta$ is the associated graph Laplacian, $\sigma>1$, $v>0$, and $w$ is a real-valued potential. The potential term is handled by means of a positive solution $H$ of $\Delta H+wH=0$, which transforms the operator $\Delta+w$ into the $H$-Laplacian associated with a new weighted graph. Our main nonexistence criterion is formulated directly in terms of cut-off functions and the regions where their $H$-Laplacian is controlled. Unlike metric criteria based on pseudo-metric annuli, our formulation determines the capacitary sets from the support of the $H$-Laplacian estimates for the cut-off functions. We provide an example showing that our result applies in situations not covered by previous nonexistence criteria based on structural lower bounds or pseudo-metric annular volume estimates. We also show that the growth exponent in our capacitary condition is sharp by constructing an example for which the condition fails by an arbitrary power $R^\varepsilon$, while a positive nontrivial solution exists.

math.AP↗

arXiv2026

Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain

We study the existence and nonexistence of weak solutions to an inhomogeneous semilinear biharmonic heat equation in an exterior domain, involving a singular Hardy--Rellich potential, a weighted nonlinearity $|x|^{\sigma}|u|^{p}$, and a positive source term $f(x)$. We identify two distinct critical regimes governing the behavior of solutions. More precisely, we first determine a Fujita-type critical exponent that separates nonexistence from existence. We then show that, in the supercritical range, a second critical exponent arises in terms of the decay exponent of the source, in the sense of Lee and Ni. Our results extend the recent work \cite{Tobakhanov} by considering a singular Hardy--Rellich potential and a weighted nonlinearity, leading to a different critical behavior.

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arXiv2026

Semilinear Heat Inequalities with a Hardy-Type Potential in an Exterior Geodesic Domain on $\mathbb{S}^N$

We study an inhomogeneous semilinear heat inequality on the unit sphere $\mathbb S^N$, $N\ge3$, in an exterior geodesic domain associated with a fixed pole. The equation involves the singular Hardy-type potential $\lambda/\sin^2 r$, where $r=d(o,x)$, and the weighted nonlinearity $(\sin r)^\alpha |u|^p$. For $\alpha>-2$ and $0<\lambda\le \lambda^*=((N-2)/2)^2$, we prove the existence of a critical exponent $p_{\mathrm{crit}}=p_{\mathrm{crit}}(\alpha,N,\lambda)$ governing the existence and nonexistence of solutions. More precisely, we prove that no weak solution exists for any nontrivial nonnegative source in the range $p>p_{\mathrm{crit}}$, whereas classical solutions exist for some positive continuous sources in the range $1 1$. The analysis is based on the construction of radial Hardy barriers adapted to the antipodal singularity and on sharp integral estimates involving power and logarithmic cutoffs near $r=\pi$.

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arXiv2025

Fixed point results for single and multi-valued three-points contractions

In this paper, we are concerned with the study of the existence of fixed points for single and multi-valued three-points contractions. Namely, we first introduce a new class of single-valued mappings defined on a metric space equipped with three metrics. A fixed point theorem is established for such mappings. The obtained result recovers that established recently by the second author [J. Fixed Point Theory Appl. 25 (2023) 74] for the class of single-valued mappings contracting perimeters of triangles. We next extend our study by introducing the class of multivalued three points contractions. A fixed point theorem, which is a multi-valued version of that obtained in the above reference, is established. Some examples showing the validity of our obtained results are provided.

math.GN↗

arXiv2024

Fixed point results for contractions of polynomial type

We introduce two new classes of single-valued contractions of polynomial type defined on a metric space. For the first one, called the class of polynomial contractions, we establish two fixed point theorems. Namely, we first consider the case when the mapping is continuous. Next, we weaken the continuity condition. In particular, we recover Banach's fixed point theorem. The second class, called the class of almost polynomial contractions, includes the class of almost contractions introduced by Berinde [Nonlinear Analysis Forum. 9(1) (2004) 43--53]. A fixed point theorem is established for almost polynomial contractions. The obtained result generalizes that derived by Berinde in the above reference. Several examples showing that our generalizations are significant, are provided.

math.GN↗

arXiv2024

New directions in fixed point theory in $G$-metric spaces and applications to mappings contracting perimeters of triangles

We are concerned with the study of fixed points for mappings $T: X\to X$, where $(X,G)$ is a $G$-metric space in the sense of Mustafa and Sims. After the publication of the paper [Journal of Nonlinear and Convex Analysis. 7(2) (2006) 289--297] by Mustafa and Sims, a great interest was devoted to the study of fixed points in $G$-metric spaces. In 2012, the first and third authors observed that several fixed point theorems established in $G$-metric spaces are immediate consequences of known fixed point theorems in standard metric spaces. This observation demotivated the investigation of fixed points in $G$-metric spaces. In this paper, we open new directions in fixed point theory in $G$-metric spaces. Namely, we establish new versions of the Banach, Kannan and Reich fixed point theorems in $G$-metric spaces. We point out that the approach used by the first and third authors [Fixed Point Theory Appl. 2012 (2012) 1--7] is inapplicable in the present study. We also provide some interesting applications related to mappings contracting perimeters of triangles.

math.GN↗

arXiv2024

Higher-order evolution inequalities with Hardy potential on the Kor\'{a}nyi ball

We consider a higher order in (time) semilinear evolution inequality posed on the Kor\'{a}nyi ball under an inhomogeneous Dirichlet-type boundary condition. The problem involves an inverse-square potential $\lambda/|\xi|_\mathbb{H}^2$, where $\lambda \geq -(Q-2)^2/4$ and a general weight function $V$ depending on the space variable in front of the power nonlinearity. We first establish a general nonexistence result for the considered problem. Next, in the special case $V(\xi):=|\xi|_\mathbb{H}^a$, $a\in \mathbb{R}$, we prove the sharpness of our nonexistence result and show that the problem admits three different critical behaviors according to the value of the parameter $\lambda$.

math.AP↗

arXiv2023

Semilinear wave inequalities with double damping and potential terms on Riemannian Manifolds

We study a semilinear wave inequality with double damping on a complete noncompact Riemannian manifold. The considered problem involves a potential function $V$ depending on the space variable in front of the power nonlinearity and an inhomogeneous term $W$ depending on both time and space variables. Namely, we establish sufficient conditions for the nonexistence of weak solutions in both cases: $W\equiv 0$ and $W\not\equiv 0$. The obtained conditions depend on the parameters of the problem as well as the geometry of the manifold. Some special cases of manifolds, and of $V$ and $W$ are discussed in detail.

math.AP↗

arXiv2023

The role of convection in the limit shape of the critical front profile for Born-Infeld diffusion models

In this paper, we deal with models with Born-Infeld type diffusion and monostable reaction, investigating the effect of the introduction of a convection term on the limit shape of the critical front profile for vanishing diffusion. We first provide an estimate of the critical speed and then, through a careful analysis of an equivalent first-order problem, we show that different convection terms may lead either to a complete sharpening of the limit profile or to its complete regularization, presenting some related numerical simulations.

math.AP↗

arXiv2023

Higher order evolution inequalities with Hardy potential in the exterior of a half-ball

We consider semilinear higher order (in time) evolution inequalities posed in an exterior domain of the half-space $\mathbb{R}_+^N$, $N\geq 2$, and involving differential operators of the form $\mathcal{L}_\lambda =-\Delta +\lambda/|x|^2$, where $\lambda\geq -N^2/4$. A potential function of the form $|x|^\tau$, $\tau\in \mathbb{R}$, is allowed in front of the power nonlinearity. Under inhomogeneous Dirichlet-type boundary conditions, we show that the dividing line with respect to existence or nonexistence is given by a Fujita-type critical exponent that depends on $\lambda, N$ and $\tau$, but independent of the order of the time derivative.

math.AP↗

arXiv2022

Nonexistence for parabolic differential inequalities with convection terms in exterior domains

We are concerned with the nonexistence of sign-changing global weak solutions for a class of semilinear parabolic differential inequalities with convection terms in exterior domains. A weight function of the form $t^\alpha |x|^\sigma$ is considered in front of the power nonlinearity. Two types of non-homogeneous boundary conditions are investigated: Neumann-type and Dirichlet-type boundary conditions. Using a unified approach, for each case, we establish sufficient criteria for the nonexistence of global weak solutions. When $\alpha=0$, the critical exponent in the sense of Fujita is obtained. This exponent is bigger than that found previously by Zheng and Wang (2008) in the case of homogeneous Neumann and Dirichlet boundary conditions.

math.AP↗

arXiv2022

On the absence of global weak solutions for a nonlinear time-fractional Schr\"odinger equation

In this paper, an initial value problem for a nonlinear time-fractional Schr\"odinger equation with a singular logarithmic potential term is investigated. The considered problem involves the left/forward Hadamard-Caputo fractional derivative with respect to the time variable. Using the test function method with a judicious choice of the test function, we obtain sufficient criteria for the absence of global weak solutions.

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arXiv2020

Liouville-type theorems for sign-changing solutions to nonlocal elliptic inequalities and systems with variable-exponent nonlinearities

We consider the fractional elliptic inequality with variable-exponent nonlinearity $$ (-\Delta)^{\frac{\alpha}{2}} u+\lambda\, \Delta u \geq |u|^{p(x)}, \quad x\in\mathbb{R}^N, $$ where $N\geq 1$, $\alpha\in (0,2)$, $\lambda\in\mathbb{R}$ is a constant, $p: \mathbb{R}^N\to (1,\infty)$ is a measurable function, and $(-\Delta)^{\frac{\alpha}{2}}$ is the fractional Laplacian operator of order $\frac{\alpha}{2}$. A Liouville-type theorem is established for the considered problem. Namely, we obtain sufficient conditions under which the only weak solution is the trivial one. Next, we extend our study to systems of fractional elliptic inequalities with variable-exponent nonlinearities. Besides the consideration of variable-exponent nonlinearities, the novelty of this work consists in investigating sign-changing solutions to the considered problems. Namely, to the best of our knowledge, only nonexistence results of positive solutions to fractional elliptic problems were invetigated previously. Our approach is based on the nonlinear capacity method combined with a pointwise estimate of the fractional Laplacian of some test functions, which was derived by Fujiwara (2018) (see also Dao and Reissig (2019)). Note that the standard nonlinear capacity method cannot be applied to the considered problems due to the change of sign of solutions.

math.AP↗

arXiv2020

Blow-up and global existence for semilinear parabolic systems with space-time forcing terms

We investigate the local existence, finite time blow-up and global existence of sign-changing solutions to the inhomogeneous parabolic system with space-time forcing terms $$ u_t-\Delta u =|v|^{p}+t^\sigma w_1(x),\,\, v_t-\Delta v =|u|^{q}+t^\gamma w_2(x),\,\, (u(0,x),v(0,x))=(u_0(x),v_0(x)), $$ where $t>0$, $x\in \mathbb{R}^N$, $N\geq 1$, $p,q>1$, $\sigma,\gamma>-1$, $\sigma,\gamma\neq0$, $w_1,w_2\not\equiv0$, and $u_0,v_0\in C_0(\mathbb{R}^N)$. For the finite time blow-up, two cases are discussed under the conditions $w_i\in L^1(\mathbb{R}^N)$ and $\int_{\mathbb{R}^N} w_i(x)\,dx>0$, $i=1,2$. Namely, if $\sigma>0$ or $\gamma>0$, we show that the (mild) solution $(u,v)$ to the considered system blows up in finite time, while if $\sigma,\gamma\in(-1,0)$, then a finite time blow-up occurs when $\frac{N}{2}< \max\left\{\frac{(\sigma+1)(pq-1)+p+1}{pq-1},\frac{(\gamma+1)(pq-1)+q+1}{pq-1}\right\}$. Moreover, if $\frac{N}{2}\geq \max\left\{\frac{(\sigma+1)(pq-1)+p+1}{pq-1},\frac{(\gamma+1)(pq-1)+q+1}{pq-1}\right\}$, $p>\frac{\sigma}{\gamma}$ and $q>\frac{\gamma}{\sigma}$, we show that the solution is global for suitable initial values and $w_i$, $i=1,2$.

math.AP↗

arXiv2019

Critical behavior for a semilinear parabolic equation with forcing term depending of time and space

We investigate the large-time behavior of the sign-changing solution of the inhomogeneous semilinear heat equation with a forcing term depending of time and space. we identify the critical exponent for this problem, which separates the nonexistence/existence of global-in-time solutions, and show the discontinuity of this critical exponent at the point which the inhomogeneous term becomes independent of time variable.

math.AP↗

arXiv2019

Nonexistence results for a higher-order evolution equation with an inhomogeneous term depending on time and space

We consider a higher-order evolution equation with an inhomogeneous term depending on time and space. We first derive a general criterion for the nonexistence of weak solutions. Next, we study the particular case when the inhomogeneity depends only on space. In that case, we obtain the first critical exponent in the sense of Fujita, as well as the second critical exponent in the sense of Lee and Ni.

math.AP↗

arXiv2019

Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities

We consider the nonlinear heat equation $u_t-\Delta u =|u|^p+b |\nabla u|^q$ in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$. First, we focus our attention on positive solutions and obtain an optimal Fujita-type result: any positive solution blows up in finite time if $p\leq 1+\frac{2}{n}$ or $q\leq 1+\frac{1}{n+1}$, while global classical positive solutions exist for suitably small initial data when $p>1+\frac{2}{n}$ and $q> 1+\frac{1}{n+1}$. Although finite time blow-up cannot be produced by the gradient term alone and should be considered as an effect of the source term $|u|^p$, this result shows that the gradient term induces an interesting phenomenon of discontinuity of the critical Fujita exponent, jumping from $p=1+\frac{2}{n}$ to $p=\infty$ as $q$ reaches the value $1+\frac{1}{n+1}$ from above. Next, we investigate the case of sign-changing solutions and show that if $p\le 1+\frac{2}{n}$ or $0<(q-1)(np-1)\le 1$, then the solution blows up in finite time for any nontrivial initial data with nonnegative mean. Finally, a Fujita-type result, with a different critical exponent, is % also obtained for sign-changing solutions to the inhomogeneous version of this problem.

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