Asymptotic Behavior Of Blow-up Solutions Research Articles

Refined Asymptotic Behavior of Blowup Solutions to a Simplified Chemotaxis System

AbstractWe deal with a parabolic‐elliptic chemotaxis system. It is known that finite‐time blowup occurs for a large class of initial data. However, there have been no results on exact blowup rate or detailed blowup behavior except a special radial solution given just formally in [13] and rigorously in [19, 10, 9]. Our aim is to show that for all radial blowup solutions, their blowup rate, and blowup profile related to mass concentration are the same as those of the special solution. This implies that there is exactly one blowup behavior including blowup rate in radial case . © 2020 Wiley Periodicals LLC.

Blow-up analysis for Neri's mean field equation in 2D-turbulence

In this paper we study the blow-up analysis for some mean field equation on point vortices which is derived by C. Neri under a stochastic assumption. In particular, we derive some estimate which is the asymptotic behavior of blow-up solutions near the blow-up points. To obtain this result we shall employ the new scaling argument for blow-up solutions of Neri's mean field equation. Moreover, we also study the power type elliptic equation in higher dimension case related to mean field equation and derive such estimate.

Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy

This paper considers the Cauchy problem for a nonlinear Klein-Gordon system with damping terms. In the existing works, the solution with low and critical initial energy was studied. We extend the previous results on following three aspects. Firstly, we consider the vacuum isolating phenomenon of solution under initial energy $ E(0)\leq0 $. We find that the corresponding vacuum region is an ball and it expands to whole phase space as $ E(0) $ decays to $ -\infty $. Secondly, we discuss the asymptotic behavior of blow-up solution and prove that the solution grows exponentially. The growth speed is estimated especially. Finally, the solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the solution exists globally and blows up in finite time are given, respectively.

On Kiguradze’s problem on power-law asymptotic behavior of blow-up solutions to Emden–Fowler type differential equations

Abstract For higher-order Emden–Fowler type equations with regular nonlinearity, the asymptotic behavior of their blow-up solutions is investigated. It is proved that for weakly superlinear differential equations, all such solutions have power-law asymptotic behavior.

The exact asymptotic behavior of blow-up solutions to a highly degenerate elliptic problem

In this paper, by constructing new comparison functions, we mainly study the boundary behavior of solutions to boundary blow-up elliptic problems for more general nonlinearities f (which may be rapidly varying at infinity) $\triangle_{\infty} u =b(x)f(u)$ , $x\in \Omega $ , $u|_{\partial \Omega }=+\infty$ , where Ω is a bounded domain with smooth boundary in $\mathbb{R}^{N}$ , and $b \in C(\bar{\Omega })$ which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary.

Blow-up rate for radially symmetric solutions of some parabolic equations with nonlinear boundary conditions

We consider the following parabolic equations with nonlinear boundary conditions: ut=Δu−a|u|p−1u in B1×(0,T), ∂νu=|u|q−1u on ∂B1×(0,T), where p,q>1, a⩾0 and B1 is the unit ball. We study the blow-up rate of radial symmetric solutions u(r,t) without any assumptions on the initial data. Furthermore we show the uniqueness of positive solutions of one dimensional backward self-similar solutions. As a consequence, we can determine the asymptotic behavior of blow-up solutions near the blow-up time.

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type −Δpu = a(x)um−b(x)f(u) with p > 1 and 0 1.

Asymptotic Behavior of Type I Blowup Solutions to a Parabolic-Elliptic System of Drift–Diffusion Type

We consider a Cauchy problem for a parabolic-elliptic system of drift–diffusion type. The problem is formally of the form $$ U_t = \nabla \cdot (\nabla U-U \nabla (-\Delta)^{-1}U). $$ This system describes a mass-conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis. Our concern is the asymptotic behavior of blowup solutions when the blowup is type I, in the sense that its blowup rate is the same as the corresponding ordinary differential equation y t = y 2 (up to a multiple constant). It is shown that all type I blowup is asymptotically (backward) self-similar, provided that the solution is radial, nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three.

Existence and asymptotic behavior of blow-up solutions to a class of [formula omitted]-Laplacian problems

This paper investigates the problem { − Δ p ( x ) u + f ( x , u ) = 0 in Ω , u ( x ) → + ∞ as d ( x , ∂ Ω ) → 0 , where − Δ p ( x ) u = − div ( | ∇ u | p ( x ) − 2 ∇ u ) is called p ( x ) -Laplacian. The existence of blow-up solutions is discussed, and the singularity of blow-up solutions is given.

Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations

Given a bounded domain Ω we consider local weak blow-up solutions to the equation Δ p u= g( x) f( u) on Ω. The non-linearity f is a non-negative non-decreasing function and the weight g is a non-negative continuous function on Ω which is allowed to be unbounded on Ω. We show that if Δ p w=− g( x) in the weak sense for some w∈W 1,p 0(Ω) and f satisfies a generalized Keller–Osserman condition, then the equation Δ p u= g( x) f( u) admits a non-negative local weak solution u∈W 1,p loc (Ω)∩C(Ω) such that u( x)→∞ as x→∂Ω. Asymptotic boundary estimates of such blow-up solutions will also be investigated.

The uniqueness of nonnegative C1-solution for nonlinear differential equations

This paper investigates the uniqueness of solutions with compact supports of a boundary value problem which comes from the study of asymptotic behavior of blow-up solution of the degenerate parabolic system. We obtain the estimates of supports and some numerical results are given.

Asymptotic Behavior of Blowup Solutions of a Parabolic Equation with the $p$-Laplacian

We consider the blowup problem for u_t = Δ_pu + ∣ u ∣^{p–2}u ( x\in Ω , t > 0 ) under the Dirichlet boundary condition and p > 2 . We derive sufficient conditions on blowing up of solutions. In particular, it is shown that every non-negative and non-zero solution blows up in a finite time if the domain Ω is large enough. Moreover, we show that every blowup solution behaves asymptotically like a self-similar solution near the blowup time. The Rayleigh type quotient introduced in Lemma A plays an important role throughout this paper.

The Asymptotic Behavior of Blowup Solution of Localized Nonlinear Equation

In this paper we are concerned with the growth rate of blowup solution to the equation[formula]Using self-similar solution technique and comparison method, we obtain the growth rate of blowup solution and observe that the boundary-layer phenomena occurs.