27 research outputs found
Global behavior of nonlocal in time reaction-diffusion equations
The present paper considers the Cauchy-Dirichlet problem for the
time-nonlocal reaction-diffusion equation where is a locally Lipschitz
function, is a linear operator. This model arises when studying
the processes of anomalous and ultraslow diffusions. Results regarding the
local and global existence, decay estimates, and blow-up of solutions are
obtained. The obtained results provide partial answers to some open questions
posed by Gal and Varma (2020), as well as Luchko and Yamamoto (2016).
Furthermore, possible quasi-linear extensions of the obtained results are
discussed, and some open questions are presented.Comment: 16 page
Critical exponents for the -Laplace heat equations with combined nonlinearities
This paper studies the large-time behavior of solutions to the quasilinear
inhomogeneous parabolic equation with combined nonlinearities. This equation is
a natural extension of the heat equations with combined nonlinearities
considered by Jleli-Samet-Souplet (Proc AMS, 2020). Firstly, we focus on an
interesting phenomenon of discontinuity of the critical exponents. In
particular, we will fill the gap in the results of Jleli-Samet-Souplet for the
critical case. We are also interested in the influence of the forcing term on
the critical behavior of the considered problem, so we will define another
critical exponent depending on the forcing term.Comment: 12 page
Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
In this paper we obtain new estimates of the Hadamard fractional derivatives
of a function at its extreme points. The extremum principle is then applied to
show that the initial-boundary-value problem for linear and nonlinear
time-fractional diffusion equations possesses at most one classical solution
and this solution depends continuously on the initial and boundary conditions.
The extremum principle for an elliptic equation with a fractional Hadamard
derivative is also proved
Some Hermite-Hadamard type inequalities in the class of hyperbolic p-convex functions
In this paper, obtained some new class of Hermite-Hadamard and
Hermite-Hadamard-Fejer type inequalities via fractional integrals for the
p-hyperbolic convex functions. It is shown that such inequalities are simple
consequences of Hermite-Hadamard-Fejer inequality for the p-hyperbolic convex
function.Comment: 11 page
On inhomogeneous exterior Robin problems with critical nonlinearities
The paper studies the large-time behavior of solutions to the Robin problem
for PDEs with critical nonlinearities. For the considered problems,
nonexistence results are obtained, which complements the interesting recent
results by Ikeda et al. [J. Differential Equations, 269 (2020), no. 1,
563-594], where critical cases were left open. Moreover, our results provide
partially answers to some other open questions previously posed by Zhang [Proc.
Roy. Soc. Edinburgh Sect. A, 131 (2001), no. 2, 451-475] and Jleli-Samet
[Nonlinear Anal., 178 (2019), 348-365].Comment: 16 page
Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions
In this paper, a maximum principle for the one-dimensional sub-diffusion
equation with Atangana-Baleanu fractional derivative is formulated and proved.
The proof of the maximum principle is based on an extremum principle for the
Atangana-Baleanu fractional derivative that is given in the paper, too. The
maximum principle is then applied to show that the initial-boundary-value
problem for the linear and nonlinear time-fractional diffusion equations
possesses at most one classical solution and this solution continuously depends
on the initial and boundary conditions.Comment: to appear in Applied Mathematics Letter
Decay estimates for the time-fractional evolution equations with time-dependent coefficients
In this paper, the initial-boundary value problems for the time-fractional
degenerate evolution equations are considered. Firstly, in the linear case, we
obtain the optimal rates of decay estimates of the solutions. The decay
estimates are also established for the time-fractional evolution equations with
nonlinear operators such as: p-Laplacian, the porous medium operator,
degenerate operator, mean curvature operator, and Kirchhoff operator. At the
end, some applications of the obtained results are given to derive the decay
estimates of global solutions for the time-fractional Fisher-KPP-type equation
and the time-fractional porous medium equation with the nonlinear source.Comment: 23 pages. The previous version of the paper has been edited according
to the comments of the reviewer
Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte Type Inequalities for Convex Functions via Fractional Integrals
The aim of this paper is to establish Hermite-Hadamard,
Hermite-Hadamard-Fej\'er, Dragomir-Agarwal and Pachpatte type inequalities for
new fractional integral operators with exponential kernel. These results allow
us to obtain a new class of functional inequalities which generalizes known
inequalities involving convex functions. Furthermore, the obtained results may
act as a useful source of inspiration for future research in convex analysis
and related optimization fields.Comment: 14 pages, to appear in Journal of Computational and Applied
Mathematic