Classification of Blow-up Solutions for a Nonlinear Parabolic Partial differential Equation

摘要

Classification by types of blow-up rates which is the same as or faster than ones of self similar solutions with the blow-up time T is available for nonlinear parabolic equations, such as (partial deriv)u/(partial deriv)t = Δu+u°. However, when we know a blow-up solution, u, but do not have the precise value of the blow-up time T, it is very difficult to classify blow-up solutions. This is one of disadvantages for investigating concrete information of asymptotic behavior of blow-up solutions. Indeed, there are many cases in the vanishing viscosity method and numerical studies that the blow-up time of u_s or its approximation is known but one of u is not given exactly. Our purpose is to classify solutions without "backward" self similar solutions.