The critical exponents of parabolic equations and blow-up in Rn

1998 ◽  
Vol 128 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Yuan-wei Qi
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Initial Value ◽  
The Cauchy Problem

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

scholarly journals Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Yusuke Yamauchi
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Life Span ◽  
Recent Result ◽  
Positive Solutions ◽  
Parabolic Equations ◽  
Blow Up ◽  
Parabolic Problems ◽  
Survey Paper ◽  
The Cauchy Problem

Since 1960's, the blow-up phenomena for the Fujita type parabolic equation have been investigated by many researchers. In this survey paper, we discuss various results on the life span of positive solutions for several superlinear parabolic problems. In the last section, we introduce a recent result by the author.

Blow-up of solutions of a quasilinear parabolic equation

2012 ◽  
Vol 142 (2) ◽  
pp. 425-448
Author(s):  
Ryuichi Suzuki ◽  
Noriaki Umeda
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Positive Function ◽  
Quasilinear Parabolic Equations ◽  
The Cauchy Problem ◽  
Image Position ◽  
Quasilinear Parabolic

We consider non-negative solutions of the Cauchy problem for quasilinear parabolic equations ut = Δum + f(u), where m > 1 and f(ξ) is a positive function in ξ > 0 satisfying f(0) = 0 and a blow-up conditionWe show that if ξm+2/N /(−log ξ)β = O(f(ξ)) as ξ ↓ 0 for some 0 < β < 2/(mN + 2), one of the following holds: (i) all non-trivial solutions blow up in finite time; (ii) every non-trivial solution with an initial datum u0 having compact support exists globally in time and grows up to ∞ as t → ∞: limtt→∞ inf|x|<Ru(x, t) = ∞ for any R > 0. Moreover, we give a condition on f such that (i) holds, and show the existence of f such that (ii) holds.

scholarly journals Blow-Up Criteria for the Modified Novikov Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Caochuan Ma ◽  
Wujun Lv
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Finite Time ◽  
Blow Up ◽  
Solution Blowing ◽  
Blowing Up ◽  
The Cauchy Problem ◽  
Novikov Equation

We investigate the Cauchy problem for the modified Novikov equation. We establish blow-up criteria on the initial data to guarantee the corresponding solution blowing up in finite time.

scholarly journals Asymptotic Properties of Solutions to the Cauchy Problem for Degenerate Parabolic Equations with Inhomogeneous Density on Manifolds

2021 ◽  
Author(s):  
Daniele Andreucci ◽  
Anatoli F. Tedeev
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Degenerate Parabolic Equations ◽  
Isoperimetric Function ◽  
Finite Speed Of Propagation ◽  
The Cauchy Problem ◽  
Inhomogeneous Density ◽  
Degenerate Parabolic

AbstractWe consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.

scholarly journals Blow-up for the sixth-order multidimensional generalized Boussinesq equation with arbitrarily high initial energy

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jianghao Hao ◽  
Aiyuan Gao
Keyword(s):  
Cauchy Problem ◽  
Fourier Transform ◽  
Finite Time ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Initial Energy ◽  
Sixth Order ◽  
The Cauchy Problem ◽  
Generalized Boussinesq Equation

AbstractIn this paper, we consider the Cauchy problem for the sixth-order multidimensional generalized Boussinesq equation with double damping terms. By using the improved convexity method combined with Fourier transform, we show the finite time blow-up of solution with arbitrarily high initial energy.

scholarly journals A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion

2020 ◽  
Vol 18 (1) ◽  
pp. 194-203 ◽  
Author(s):  
Zaiyun Zhang ◽  
Limei Li ◽  
Chunhua Fang ◽  
Fan He ◽  
Chuangxia Huang ◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Type Equation ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Math Phys ◽  
Blow Up Criterion

Abstract In this paper, we investigate the Cauchy problem for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion. Furthermore, we establish the blow-up result of the positive solutions in finite time under certain conditions on the initial datum. This result complements the early one in the literature, such as [E. Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation, J. Math. Phys. 54 (2013), no. 9, 092703, DOI 10.1063/1.4820786] and [Z.Y. Zhang, J.H. Huang, and M.B. Sun, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited, J. Math. Phys. 56 (2015), no. 9, 092701, DOI 10.1063/1.4930198].

scholarly journals On the Cauchy Problem of a Quasilinear Degenerate Parabolic Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-12
Author(s):  
Zongqi Liang ◽  
Huashui Zhan
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Classical Solution ◽  
Finite Time ◽  
Numerical Experiments ◽  
Degenerate Parabolic Equation ◽  
Blow Up ◽  
Strong Solutions ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

By Oleinik's line method, we study the existence and the uniqueness of the classical solution of the Cauchy problem for the following equation in[0,T]×R2:∂xxu+u∂yu−∂tu=f(⋅,u), provided thatTis suitable small. Results of numerical experiments are reported to demonstrate that the strong solutions of the above equation may blow up in finite time.

BREZIS–MERLE INEQUALITIES AND APPLICATION TO THE GLOBAL EXISTENCE OF THE CAUCHY PROBLEM OF THE KELLER–SEGEL SYSTEM

2011 ◽  
Vol 13 (05) ◽  
pp. 795-812 ◽  
Author(s):  
TOSHITAKA NAGAI ◽  
TAKAYOSHI OGAWA
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Parabolic Equations ◽  
Parabolic Systems ◽  
Unique Global Solution ◽  
Drift Diffusion ◽  
Nonlinear Parabolic ◽  
Elliptic And Parabolic Equations ◽  
The Cauchy Problem ◽  
Two Space Dimensions

We discuss the existence of the global solution for two types of nonlinear parabolic systems called the Keller–Segel equation and attractive drift–diffusion equation in two space dimensions. We show that the system admits a unique global solution in [Formula: see text]. The proof is based upon the Brezis–Merle type inequalities of the elliptic and parabolic equations. The proof can be applied to the Cauchy problem which is describing the self-interacting system.

LOCAL WELL POSEDNESS OF CAUCHY PROBLEM FOR VISCOUS DIFFUSION EQUATIONS

2009 ◽  
Vol 20 (04) ◽  
pp. 509-519
Author(s):  
YACHENG LIU ◽  
RUNZHANG XU
Keyword(s):  
Cauchy Problem ◽  
Integral Equations ◽  
Finite Time ◽  
Blow Up ◽  
Diffusion Equations ◽  
Well Posedness ◽  
The Cauchy Problem

In this paper, we study the Cauchy problem of multi-dimensional viscous diffusion equations. By using an equivalent integral equations, we get the existence of local Wk,p solutions. And we prove the finite time blow up of solutions under appropriate conditions.

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

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