Our motivation is a mathematical model describing the spatial propagation of an epidemic disease through a population. In this model, the pathogen diversity is structured into two clusters and then the population is divided into eight classes which permits to distinguish between the infected/uninfected population with respect to clusters. In this paper, we prove the weak and the global existence results of the solutions for the considered reaction-diffusion system with Neumann boundary. Next, mathematical Turing formulation and numerical simulations are introduced to show the pattern formation for such systems. [ABSTRACT FROM AUTHOR]
*INITIAL value problems, *EQUATIONS, *NUMERICAL analysis, *ALGEBRA, *BOUNDARY value problems
Abstract
A class of periodic initial value problems for two-dimensional Newton-Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved. [ABSTRACT FROM AUTHOR]
*BOUNDARY value problems, *BLOWING up (Algebraic geometry), *PARABOLIC differential equations, *NONLINEAR systems, *NUMERICAL analysis
Abstract
In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation ut−△ut−△u−div(|∇u|2q∇u)=up which was studied by Peng et al. (Appl. Math. Lett. 56:17-22, 2016), where the blow-up phenomena and the lifespan for the initial energy J(u0)<0 were obtained. We establish the finite time blow-up of the solution for the initial data at arbitrary energy level and the lifespan of the blow-up solution. Furthermore, as a product, we obtain the blow-up rate and refine the lifespan when J(u0)<0. [ABSTRACT FROM AUTHOR]