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Bessem Samet

Publications and source records attributed to Bessem Samet.

28 records · Page 2Linked to original sources
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 $u_t-Δu =|u|^p+b |\nabla u|^q$ 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.

math.AP↗
arXiv2019

Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains

We consider blow-up results for a system of inhomogeneous wave inequalities in exterior domains. We will handle three type boundary conditions: Dirichlet type, Neumann type and mixed boundary conditions. We use a unified approach to show the optimal criteria of Fujita type for each case. Our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation. We provide many new results and close some open questions.

math.AP↗
arXiv2018

A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel-function. Several properties related to this new operator, like inversion rules, integration by parts, etc. are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann-Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.

math.CA↗
arXiv2018

On a new generalization of metric spaces

In this paper, we introduce the $\mathcal{F}$-metric space concept, which generalizes the metric space notion. We define a natural topology $τ_{\mathcal{F}}$ in such spaces and we study their topological properties. Moreover, we establish a new version of the Banach contraction principle in the setting of $\mathcal{F}$-metric spaces. Several examples are presented to illustrate our study.

math.GN↗
arXiv2018

On new types of fractional operators and applications

We introduce two kinds of fractional integral operators; the one is defined via the exponential-integral function $$ E_1(x)=\int_x^\infty \frac{e^{-t}}{t}\,dt,\quad x>0, $$ and the other is defined via the special function $$ \mathcal{S}(x)=e^{-x} \int_0^\infty \frac{x^{s-1}}{Γ(s)}\,ds,\quad x>0. $$ We establish different properties of these operators, and we study the relationship between the fractional integrals of first kind and the fractional integrals of second kind. Next, we introduce a new concept of fractional derivative of order $α>0$, which is defined via the fractional integral of first kind. Using an approximate identity argument, we show that the introduced fractional derivative converges to the standard derivative in $L^1$ space, as $α\to 0^+$. Several other properties are studied, like fractional integration by parts, the relationship between this fractional derivative and the fractional integral of second kind, etc. As an application, we consider a new fractional model of the relaxation equation, we establish an existence and uniqueness result for this model, and provide an iterative algorithm that converges to the solution.

math.CA↗
arXiv2012

Existence of positive solutions for generalized Lyapunov equations via a coupled fixed point theorem

We consider the generalized continuous-time Lyapunov equation: $$ A^*XB + B^*XA =-Q, $$ where $Q$ is an $N\times N$ Hermitian positive definite matrix and $A,B$ are arbitrary $N\times N$ matrices. Under some conditions, using the coupled fixed point theorem of Bhaskar and Lakshmikantham, we establish the existence and uniqueness of Hermitian positive definite solution for such equation. Moreover, we provide an iteration method to find convergent sequences which converge to the solution if one exists.

math.NA↗