Non-existence of radial backward self-similar blow-up solutions with sign change

2011 ◽  
Vol 141 (4) ◽  
pp. 825-834
Author(s):  
Noriko Mizoguchi
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Blow Up ◽  
Similar Solution ◽  
Semilinear Heat Equation ◽  
Sign Change ◽  
Blowing Up ◽  
Self Similar Solution ◽  
Self Similar ◽  
Image Position

We consider a Cauchy problem for a semilinear heat equationwith p > 1. If u(x, t) = (T − t)−1/(p−1)ϕ((T − t)−1/2x) for x ∈ ℝN and t ∈ [0, T),where ϕ ∈ L∞(ℝN) is a solution not identically equal to zero ofthen u is called a backward self-similar solution blowing up at t = T. We show that, for all p > 1, there exists no radial sign-changing solution of (E) which belongs to L∞(ℝN). This implies the non-existence of radial backward self-similar solution with sign change blowing up in finite time.

On backward self-similar blow-up solutions to a supercritical semilinear heat equation

2010 ◽  
Vol 140 (4) ◽  
pp. 821-831 ◽  
Author(s):  
Noriko Mizoguchi
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Blow Up ◽  
Radial Solution ◽  
Similar Solution ◽  
Semilinear Heat Equation ◽  
Blowing Up ◽  
Self Similar Solution ◽  
Self Similar ◽  
Image Position

We are concerned with a Cauchy problem for the semilinear heat equationthen u is called a backward self-similar solution blowing up at t = T. Let pS and pL be the Sobolev and the Lepin exponents, respectively. It was shown by Mizoguchi (J. Funct. Analysis257 (2009), 2911–2937) that k ≡ (p − 1)−1/(p−1) is a unique regular radial solution of (P) if p > pL. We prove that it remains valid for p = pL. We also show the uniqueness of singular radial solution of (P) for p > pS. These imply that the structure of radial backward self-similar blow-up solutions is quite simple for p ≥ pL.

Minimal blow-up asymptotics of quasilinear heat equations

2001 ◽  
Vol 131 (6) ◽  
pp. 1297-1321
Author(s):  
M. Chaves ◽  
Victor A. Galaktionov
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Basin Of Attraction ◽  
Boundary Function ◽  
Similar Solution ◽  
Heat Equations ◽  
Blowing Up ◽  
Self Similar Solution ◽  
Self Similar

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) > 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 .

On blow-up and degeneracy for the semilinear heat equation with source

1990 ◽  
Vol 115 (1-2) ◽  
pp. 19-24 ◽  
Author(s):  
V. A. Galaktionov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Semilinear Parabolic Equation ◽  
Quasilinear Parabolic Equations ◽  
Open Problems ◽  
Semilinear Heat Equation ◽  
Self Similar Solution ◽  
Self Similar ◽  
Semilinear Parabolic ◽  
Hamilton Jacobi Equation

SynopsisThe asymptotic behaviour of the solution of the semilinear parabolic equation ut = uxx + (1 + u)ln2(l + u) for t > 0, x ∊[−π, π ], ux(t, ± π) = 0 for t > 0 and u(0, x) = u0(x) ≧ 0 in [−π, π], which blows up at a finite time T0, is investigated. It is proved that for some two-parametric set of initial functions u0 the behaviour of u(t, x) near t = T0 is described by the approximate self-similar solution va(t, x) = exp {(T0 −t)−1 cos2 (x/2)} − 1, satisfying the first order nonlinear Hamilton–Jacobi equation vt, = (vx)2 /(1 + v) + (1 + v) ln2 (1 + v). Some open problems of degeneracy near a finite blow-up time for other semilinear or quasilinear parabolic equations with source ut, = Δu + (1 + u) lnβ (1 + u) (β >1), ut, = Δu + uβ(β > l), ut = Δu + eu; ut = ∇. (lnσ(1 + u)∇u)+ (1 + u)lnβ(1 + u) (σ > 0, β > 1) are discussed.

A semilinear heat equation with singular initial data

1998 ◽  
Vol 128 (4) ◽  
pp. 745-758 ◽  
Author(s):  
Minkyu Kwak
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Decay Rate ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Semilinear Heat Equation ◽  
Singular Initial Data ◽  
Self Similar ◽  
Image Position

We first prove existence and uniqueness of non-negative solutions of the equationin in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the formwhere g = ga satisfiesAfter uniqueness is proved, the asymptotic behaviour of solutions ofis studied. In particular, we show thatThe case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

Convergence of blow-up solutions of nonlinear heat equations in the supercritical case

1999 ◽  
Vol 129 (6) ◽  
pp. 1197-1227 ◽  
Author(s):  
J. Matos
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Convergence Theorem ◽  
Nonlinear Parabolic Equation ◽  
Blow Up ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
Nonlinear Parabolic ◽  
Nonlinear Heat Equations ◽  
Self Similar

In this paper, we study the blow-up behaviour of the radially symmetric non-negative solutions u of the semilinear heat equation with supercritical power nonlinearity up (that is, (N – 2)p> N + 2). We prove the existence of non-trivial self-similar blow-up patterns of u around the blow-up point x = 0. This result follows from a convergence theorem for a nonlinear parabolic equation associated to the initial one after rescaling by similarity variables.

Influence of the Hardy potential in a semilinear heat equation

2009 ◽  
Vol 139 (5) ◽  
pp. 897-926 ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Ireneo Peral ◽  
Ana Primo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Critical Power ◽  
Regular Domain ◽  
Hardy Potential ◽  
Semilinear Heat Equation ◽  
The Cauchy Problem ◽  
Image Position ◽  
Class Of Functions ◽  
Bounded Regular Domain

This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problemwhere Ω⊂ℝN, N≥3, is a bounded regular domain such that 0∈Ω, p>1, and u0≥0, f≥0 are in a suitable class of functions.There is a great difference between this result and the heat equation (λ=0); indeed, if λ>0, there exists a critical exponent p+(λ) such that for p≥p+(λ) there is no solution for any non-trivial initial datum.The Cauchy problem, Ω=ℝN, is also analysed for 1<p<+(λ). We find the same phenomenon about the critical power p+(λ) as above. Moreover, there exists a Fujita-type exponent, F(λ), in the sense that, independently of the initial datum, for 1<p<F(λ), any solution blows up in a finite time. Moreover, F(λ)>1+2/N, which is the Fujita exponent for the heat equation (λ=0).

Blow-up for quasilinear heat equations with critical Fujita's exponents

1994 ◽  
Vol 124 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Victor A. Galaktionov
Keyword(s):  
Heat Equation ◽  
Critical Exponent ◽  
Critical Case ◽  
Source Term ◽  
Blow Up ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
The Cauchy Problem ◽  
Fixed Constant ◽  
Image Position

We consider the Cauchy problem for the quasilinear heat equationwhere σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivitywith σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.

scholarly journals Global blow-up for a semilinear heat equation on a subspace

2015 ◽  
Vol 145 (5) ◽  
pp. 893-923 ◽  
Author(s):  
C. J. Budd ◽  
J. W. Dold ◽  
V. A. Galaktionov
Keyword(s):  
Heat Equation ◽  
Initial Function ◽  
Blow Up ◽  
Neumann Boundary ◽  
Compact Set ◽  
Sharp Estimates ◽  
Semilinear Heat Equation ◽  
Non Local ◽  
Image Position ◽  
Local Term

We study the asymptotic behaviour as t → T–, near a finite blow-up time T > 0, of decreasing-in-x solutions to the following semilinear heat equation with a non-local term:with Neumann boundary conditions and strictly decreasing initial function u0(x) with zero mass. We prove sharp estimates for u(x, t) as t → T–, revealing a non-uniform global blow-up:uniformly on any compact set [δ, 1], δ ∈ (0, 1).

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