Boundary layer for nonlinear evolution equations with damping and diffusion

Boundary Layer Effects for the Nonlinear Evolution Equations with the Vanishing Diffusion Limit

“…which implies that ( , ) (0, ) = ( , ) (1, ) = (0, 0) , ≥ 0, (4) where , , , and are positive constants with > 1 and 0 < < 1. The limit problem of the vanishing parameter → 0 + ; we have 0 = 0, 0 = − (1 − ) 0 + 2 0 0 + 0 , 0 < < 1, > 0 (5) with the initial conditions ( 0 , 0 ) ( , 0) = ( 1 ( ) , 2 ( )) , 0 ≤ ≤ 1,…”

“…System (1) has been extensively studied by several authors in different contexts [2,3,[5][6][7][8][9], such as the well-posedness problems, the nonlinear stability problem, and optimal decay rate of the solutions. However, all the results above need to assume that all parameters are fixed constants.…”

New Expansions and Breakthroughs of the 21st Century AI Computing Life

“…Numerical calculations in the research of life science include: numerical methods of nonlinear equations, numerical methods of linear equations, matrix method, interpolation, least squares and curve fitting, surface fitting, numerical calculus, numerical methods of ordinary differential equations and partial differential equations, and error analysis in numerical calculation, etc. Through the above numerical calculations, we can provide accurate, efficient and reliable scientific calculation methods and methods for related problems in life science [8]. 4.Mathematical model method In the study of biological mathematics, we often use mathematical models to transform a complex life matter movement into a mathematical problem.…”

Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis

“…In the case of ε > 0, for the initial-boundary value problem of (1.1) with Dirichlet boundary conditions u ε (0, t) = u ε (1, t) = 0, v ε (0, t) = v ε (1, t) = 0, with the help of the methods in [25], we can prove the global unique solvability in the setting of small amplitude since the difficulties derived from the boundary effect on the viscosity εu ε xx . What's more, the boundary layer will occur and the boundary layer thickness will be shown as in [4,7,18]. These are left to our future study since these are not in accordance with our main aim in the paper, which mainly refers to large amplitude solution.…”