Nonexistence of Global Solutions of Systems of Time Fractional Differential equations posed on the Heisenberg group

We first consider the nonlinear time fractional diffusion equation D^{1+α}u+D^β u−∆_{H} u=|u|^p posed on the Heisenberg group H, where 1 < p is a positive real nimber to be specified later; D^δ_{0|t} is the Liouville-Caputo derivative of order δ. For 0 < α < 1,0 < β ≤ 1. This equation interpolates the heat equation and the wave equation with the linear damping D^β_{0|t}u. We present the Fujita exponent for blow-up. Then establish sufficient conditions ensuring non-existence of local solutions. We extend the analysis to the case of the system D^{1+α}u+D^β u−∆_{H} u=|v|^q D^{1+δ}v+D^γ v−∆_{H} v=|u|^p. Our method of proof is based on the nonlinear capacity method.

Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

Blowing-up solutions of the time-fractional dispersive equations

This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

Fujita type results for quasilinear parabolic inequalities with nonlocal terms

<p style='text-indent:20px;'>In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} &amp;u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &amp;u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u_0\in L^1_{loc}({\mathbb R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ L_{\mathcal{A}} $\end{document}</tex-math></inline-formula> denotes a weakly <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>-coercive operator, which includes as prototype the <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>-Laplacian or the generalized mean curvature operator, <inline-formula><tex-math id="M5">\begin{document}$ p,\,q&gt;0 $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M6">\begin{document}$ K\ast u^p $\end{document}</tex-math></inline-formula> stands for the standard convolution operator between a weight <inline-formula><tex-math id="M7">\begin{document}$ K&gt;0 $\end{document}</tex-math></inline-formula> satisfying suitable conditions at infinity and <inline-formula><tex-math id="M8">\begin{document}$ u^p $\end{document}</tex-math></inline-formula>. For problem <inline-formula><tex-math id="M9">\begin{document}$ (P^-) $\end{document}</tex-math></inline-formula> we obtain a Fujita type exponent while for <inline-formula><tex-math id="M10">\begin{document}$ (P^+) $\end{document}</tex-math></inline-formula> we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.</p>

Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation

In this paper, we consider the Cauchy problem for a second-order nonlinear equation with mixed fractional derivatives related to the fractional Khokhlov-Zabolotskaya equation. We prove the nonexistence of a classical local in time solution. The obtained instantaneous blow-up result is proved via the nonlinear capacity method.

On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient

This paper is concerned with the nonexistence of global solutions to fractional in time nonlinear Schrödinger equations of the form i α ∂ t α ω ( t , z ) + a 1 ( t ) Δ ω ( t , z ) + i α a 2 ( t ) ω ( t , z ) = ξ | ω ( t , z ) | p , ( t , z ) ∈ ( 0 , ∞ ) × R N , where N ≥ 1 , ξ ∈ C \ { 0 } and p > 1 , under suitable initial data. To establish our nonexistence theorem, we adopt the Pohozaev nonlinear capacity method, and consider the combined effects of absorption and dispersion terms. Further, we discuss in details some special cases of coefficient functions a 1 , a 2 ∈ L l o c 1 ( [ 0 , ∞ ) , R ) , and provide two illustrative examples.

A Fractional-Order Kinetic Battery Model of Lithium-Ion Batteries Considering a Nonlinear Capacity

Accurate battery models are integral to the battery management system and safe operation of electric vehicles. Few investigations have been conducted on the influence of current rate (C-rate) on the available capacity of the battery, for example, the kinetic battery model (KiBaM). However, the nonlinear characteristics of lithium-ion batteries (LIBs) are closer to a fractional-order dynamic system because of their electrochemical materials and properties. The application of fractional-order models to represent physical systems is timely and interesting. In this paper, a novel fractional-order KiBaM (FO-KiBaM) is proposed. The available capacity of a ternary LIB module is tested at different C-rates, and its parameter identifications are achieved by the experimental data. The results showed that the estimated errors of available capacity in the proposed FO-KiBaM were low over a wide applied current range, specifically, the mean absolute error was only 1.91%.