NONTRIVIAL EQUILIBRIUM SOLUTIONS AND GENERAL 2 STABILITY FOR STOCHASTIC EVOLUTION EQUATIONS WITH PANTOGRAPH DELAY AND TEMPERED FRACTIONAL NOISE∗

In this paper, we study the full compressible Navier--Stokes system in a bounded domain Ω⊂R3 , where the viscosity and heat conductivity depend on temperature in a power law (θb for some constant b>0 ) of Chapman--Enskog. We obtain the local existence of strong solution to the initial-boundary value problem (IBVP), which is not trivial, especially for the nonisentropic system with vacuum and temperature-dependent viscosity. There is degeneracy caused by vacuum, and there is extremely strong nonlinearity caused by variable coefficients, both of which create great difficulty for the a priori estimates, especially for the second-order estimates. First, in order to obtain closed first-order estimates, we introduce a new variable to reformulate the system into a better form and require the measure of initial vacuum domain to be sufficiently small. Second, with the help of a cut-off and straightening out technique, and the thermo-insulated boundary condition, we establish the time involved estimate for the second-order derivative of temperature, which plays a key role in closing the a priori estimates. Moreover, our local existence result holds for the cases that the viscosity and heat conductivity depend on θ with possibly different power laws (i.e., μ,λ∼θb1 , κ∼θb2 with constants b1,b2∈[0,+∞) )

Almost sure stability with general decay rate of neutral stochastic pantograph equations with Markovian switching

This paper focuses on the general decay stability of nonlinear neutral stochastic pantograph equations with Markovian switching (NSPEwMSs). Under the local Lipschitz condition and non-linear growth condition, the existence and almost sure stability with general decay of the solution for NSPEwMSs are investigated. By means of M-matrix theory, some sufficient conditions on the general decay stability are also established for NSPEwMSs

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia