Properties Of Blow-up Solutions Research Articles

Properties of blow-up solutions and their initial data for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper is concerned with blow-up solutions to the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type{ut=∇⋅(∇um−uq−1∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0 under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂RN (N≥3), m≥1, q≥2. As the basis on this study, it was recently shown that there exist radial initial data such that the corresponding solutions blow up in the case q>m+2N ([5]). In the parabolic–elliptic case Sugiyama [27] established behavior of blow-up solutions; however, behavior in the parabolic–parabolic case has not been studied. The purpose of this paper is to give many finite-time blow-up solutions and behavior of blow-up solutions in a neighborhood of blow-up time in the parabolic–parabolic case.

Dynamics of blow-up solutions for the Schr\xf6dinger\u2013Choquard equation

In this paper, we study the dynamics of blow-up solutions for the nonlinear Schrödinger–Choquard equation \t\t\tiψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$i\\psi_{t}+\\Delta \\psi =\\lambda_{1}\\vert \\psi \\vert ^{p_{1}}\\psi +\\lambda_{2}\\bigl(I _{\\alpha }\\ast \\vert \\psi \\vert ^{p_{2}}\\bigr)\\vert \\psi \\vert ^{p_{2}-2}\\psi. $$\\end{document} We first show existence of blow-up solutions and obtain a sharp threshold mass of global existence and blow-up for this equation with lambda_{1}>0, lambda_{2}<0, 0< p_{1}<frac{4}{N} and p_{2}=1+frac{2+alpha }{N}. Then we obtain some dynamical properties of blow-up solutions by the corresponding ground state of this equation with lambda_{1}=0.

Minimal blow-up asymptotics of quasilinear heat equations

We study the asymptotic properties of blow-up solutions u = u(x, t) ≥ 0 of the quasilinear heat equation , where k(u) is a smooth non-negative function, with a given blowing up regime on the boundary u(0, t) = ψ(t) &gt; 0 for t ∈ (0, 1), where ψ(t) → ∞ as t → 1−, and bounded initial data u(x, 0) ≥ 0. We classify the asymptotic properties of the solutions near the blow-up time, t → 1−, in terms of the heat conductivity coefficient k(u) and of boundary data ψ(t); both are assumed to be monotone. We describe a domain, denoted by , of minimal asymptotics corresponding to the data ψ(t) with a slow growth as t → 1− and a class of nonlinear coefficients k(u).We prove that for any problem in S11−, such a blow-up singularity is asymptotically structurally equivalent to a singularity of the heat equation ut = uxx described by its self-similar solution of the form u*(x, t) = −ln(1 − t) + g(ξ), ξ = x/(1 − t)1/2, where g solves a linear ordinary differential equation. This particular self-similar solution is structurally stable upon perturbations of the boundary function and also upon nonlinear perturbations of the heat equation with the basin of attraction .