Boundedness in a nonlinear attraction-repulsion Keller–Segel system with production and consumption

Improvements and generalizations of results concerning attraction‐repulsion chemotaxis models

“…These basic statements can be proved by standard reasoning; in particular, when $$$ h\equiv 0 $$$, they verbatim follow from Frassu et al, 1, Lemmas 4.1 and 4.2 and relation () is the well‐known mass conservation property. Conversely, in the presence of the logistic terms $$$ h $$$ as in (), some straightforward adjustments have to be considered, and the $$$ {L}^1 $$$‐bound of $$$ u $$$ is consequence of an integration of the first equation in () and an application of the Hölder inequality: precisely for $$$ {k}_{+}=\max \left\{k,0\right\} $$$ $$$$ \frac{d}{dt}{\int}_{\Omega}u={\int}_{\Omega}h(u)=k{\int}_{\Omega}u-\mu {\int}_{\Omega}{u}^{\beta}\le {k}_{+}{\int}_{\Omega}u-\frac{\mu }{{\left|\Omega \right|}^{\beta -1}}{\left({\int}_{\Omega}u\right)}^{\beta}\kern0.30em \mathrm{for}\ \mathrm{all}\kern0.4em t\in \left(0,{T}_{\mathrm{max}}\right), $$$$ and we can conclude by invoking an ODI‐comparison argument.…”

“…In our computations, beyond the above positions, some uniform bounds of $$$ {\left\Vert v\left(\cdotp, t\right)\right\Vert}_{W^{1,s}\left(\Omega \right)} $$$ are required. In this sense, the following lemma gets the most out from $$$ {L}^p $$$‐ $$$ {L}^q $$$ (parabolic) maximal regularity; this is a cornerstone, and for some small values of $$$ \alpha $$$, the succeeding $$$ {W}^{1,s} $$$‐estimates are sharper than the $$$ {W}^{1,2} $$$‐estimates derived in Frassu et al, 1,2 and therein employed.…”

“…These basic statements can be proved by standard reasoning; in particular, when h ≡ 0, they verbatim follow from Frassu et al, 1, Lemmas 4.1 and 4.2 and relation (5) is the well-known mass conservation property. Conversely, in the presence of the logistic terms h as in (4), some straightforward adjustments have to be considered, and the L 1 -bound of u is consequence of an integration of the first equation in (1) and an application of the Hölder inequality: precisely for k + = max{k, 0}…”

Oscillation of Solutions to Third-Order Nonlinear Neutral Dynamic Equations on Time Scales

“…Half-linear equations, as the classical nonlinear equations, arise in the analyses of p-Laplace equations, non-Newtonian fluid theory, porous medium problems, chemotaxis models, and so forth; see, for instance, the papers [13,14,[25][26][27] for more details. On the basis of the above discussion, we will establish integral criteria and Kamenev-type criteria (see, e.g., [15]) for the oscillation of (1) by employing a similar Riccati transformation as (2).…”

Approximate solutions of linear time-fractional differential equations

“…The interest of researchers in handling fractional differential equations has greatly increased due to their applications in the field of science and engineering. Some of these applications are found in Physics, Chemistry, Signal and Image processing, Economics, Biology and so on [4,7,18,19,27,29,30]. Recently, it was investigated that fractional differential equations are used to model real world problems in accurate way as when compared with classical order and these has motivated researchers to be looking for a way to handle these fractional order differential equations, this is because there is no precise method that will yield an exact solution for fractional differential equation, it is only approximate solutions that would be derived in solving it [2,20,23,24,36].…”