Browsing by Subject "differential inequalities"

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Access status: Open Access ,
Concavity of solutions of a 2n-th order problem with symmetry
(2013) Al Twaty, Abdulmalik; Eloe, Paul W.
In this article we apply an extension of a Leggett-Williams type fixed point theorem to a two-point boundary value problem for a $2n$-th order ordinary differential equation. The fixed point theorem employs concave and convex functionals defined on a cone in a Banach space. Inequalities that extend the notion of concavity to $2n$-th order differential inequalities are derived and employed to provide the necessary estimates. Symmetry is employed in the construction of the appropriate Banach space.
Access status: Open Access ,
Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term
(Wydawnictwa AGH, 2023) Alshehri, Aisha; Aljaber, Noha; Altamimi, Haya; Alessa, Rasha; Majdoub, Mohamed
The purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables $u_t-\Delta u=|x|^{\alpha} |u|^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},$ where $\alpha \in \mathbb{R}$, $p \gt 1$, and ${\mathtt a}(t)$ as well as ${\mathbf w}(x)$ are suitable given functions. We generalize and somehow improve earlier existing works by considering a wide class of forcing terms that includes the most common investigated example $t^\sigma\,{\mathbf w}(x)$ as a particular case. Using the test function method and some differential inequalities, we obtain sufficient criteria for the nonexistence of global weak solutions. This criterion mainly depends on the value of the limit $\lim_{t\to\infty} \frac{1}{t}\,\int_0^t\,{\mathtt a}(s)\,ds$. The main novelty lies in our treatment of the nonstandard condition on the forcing term.