This paper is concerned with establishing necessary or sufficient conditions for the existence of solutions to evolution equations with fractional derivatives in space and time. The Fujita exponent is determined. Then, these results are extended to systems of reaction-diffusion equations. Our new results shed lights on important practical questions.  2005 Elsevier Inc. All rights reserved.

In this article, we analyze a spatio-temporally nonlocal nonlinear parabolic equation. First, we validate the equation by an existence-uniqueness result. Then, we show that blowing-up solutions exist and study their time blow-up profile. Also, a result on the existence of global solutions is presented. Furthermore, we establish necessary conditions for local or global existence.

The aim of this paper is to establish Hermite-Hadamard, Hermite-Hadamard-Fejér, 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.2000 Mathematics Subject Classification. 26A33; 26D10.

Abstract. We carry out spectral analysis of one class of integral operators associated with fractional order differential equations that arise in mechanics. We establish a connection between the eigenvalues of these operators and the zeros of Mittag-Leffler type functions. We give sufficient conditions for complete nonselfadjointness and completeness of systems of the eigenvalues. We prove the existence and uniqueness of the solutions for several kinds of two-point boundary value problems of fractional differential equations with Caputo's derivatives or Riemann-Liouville Derivatives, and design the single shooting methods to solve them numerically.Chapter 1. Regional problems for differential equations of fractional order 1. The basic concepts The spectral analysis of operators of the form

We present an inequality for fractional derivatives related to the Leibniz rule that not only answers positively a conjecture raised by J. I. Diaz, T. Pierantozi, and L. Vázquez but also helps us to obtain a modern proof of the maximum principle for fractional differential equations. The inequality turns out to be versatile in nature as it can be used to obtain a priori estimates for many fractional differential problems.

We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

We investigate a backward problem for the Rayleigh‐Stokes problem, which aims to determine the initial status of some physical field such as temperature for slow diffusion from its present measurement data. This problem is well‐known to be ill‐posed because of the rapid decay of the forward process. We construct a regularized solution using the filter regularization method in the Gaussian random noise. Under some a priori assumptions on the exact solution, we establish the expectation between the exact solution and the regularized solution in the L2 and Hm norms.

We investigate the profile of the blowing up solutions to the nonlinear nonlocal system (FDS) u (t) + D α 0 + (u − u 0)(t) = |v(t)| q , t > 0, v (t) + D β 0 + (v − v 0)(t) = |u(t)| p , t > 0, where u(0) = u 0 > 0, v(0) = v 0 > 0, p > 1, q > 1 are given constants and D α 0 + and D β 0 + , 0 < α < 1, 0 < β < 1 stand for the Riemann-Liouville fractional derivatives. Our method of proof relies on comparisons of the solution to the (FDS) with solutions of the subsystems obtained from (FDS) by dropping either the usual derivatives or the fractional derivatives.