Fujita type critical exponent for a free boundary problem with spatial–temporal source

In this paper, we investigate a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source. To portray the asymptotic behavior of the solution, we first derive some sufficient conditions for finite time blowup. Furthermore, the global vanishing solution is also obtained for a class of small initial data. Finally, a sharp threshold trichotomy result is provided in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution. In particular, our results show that such a problem always possesses a Fujita type critical exponent whenever the spatial source is just equivalent to a trivial constant, or is an extreme one, such as “very negative” one in the sense of measure or integral.     

Dividend optimization for jump–diffusion model with solvency constraints

Belhaj (2010) established that a barrier strategy is optimal for the dividend problem under jump–diffusion model. However, if the optimal dividend barrier level is set too low, then the bankruptcy probability may be too high to be acceptable. This paper aims to address this issue by taking the solvency constrain into consideration. Precisely, we consider a dividend payment problem with solvency constraint under a jump–diffusion model. Using stochastic control and PIDE, we derive the optimal dividend strategy of the problem.     

A nonlocal multi-point singular fractional integro-differential problem of Lane–Emden type

In this paper, using Riemann–Liouville integral and Caputo derivative, we study a nonlinear singular integro-differential equation of Lane–Emden type with nonlocal multi-point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results.     

Global existence of solutions for a p-Laplacian equation with nonlocal Fisher–KPP type reaction terms

This work is concerned with the Neumann initial boundary value problem and Cauchy problem of a parabolic p-Laplacian equation with nonlocal Fisher–KPP type reaction terms. We establish a uniform boundedness and global existence of solutions to the equation by applying the method of multipliers and modified Moser's iteration technique for some ranges of parameters. The ranges of parameters have similar structure to that of the classical critical Fujita exponent.     

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.