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.

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.

Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data

2019 ◽  
Vol 150 (2) ◽  
pp. 789-811
Author(s):  
Yūki Naito
Keyword(s):  
Initial Data ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Heat Equations ◽  
Complicated Manner ◽  
The Cauchy Problem ◽  
Self Similar ◽  
The One ◽  
Image Position ◽  
Similar Solutions

AbstractWe consider the Cauchy problem $$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill & {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill & {x\in {\bf R}^N,} \hfill \cr } } \right.$$where N > 2, p > 1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x| → ∞, and investigate the convergence property of the global solutions to the forward self-similar solutions. We first give the precise description of the relationship between the spatial decay of initial data and the large time behaviour of solutions, and then we show the existence of solutions with a time decay rate slower than the one of self-similar solutions. We also show the existence of solutions that behave in a complicated manner.

scholarly journals Asymptotic behaviour and blow-up of some unbounded solutions for a semilinear heat equation

1996 ◽  
Vol 39 (1) ◽  
pp. 81-96
Author(s):  
D. E. Tzanetis
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Heat Equation ◽  
Unbounded Solutions ◽  
Semilinear Heat Equation ◽  
Initial Boundary Value

The initial-boundary value problem for the nonlinear heat equation u1 = Δu + λf(u) might possibly have global classical unbounded solutions, for some “critical” initial data . The asymptotic behaviour of such solutions is studied, when there exists a unique bounded steady state w(x;λ) for some values of λ We find, for radial symmetric solutions, that u*(r, t)→w(r) for any 0<r≤l but supu*(·, t) = u*(0, t)→∞, as t→∞. Furthermore, if , where is some such critical initial data, then û = u(x, t; û0) blows up in finite time provided that f grows sufficiently fast.

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 a self-similar solution for the decay of turbulent bursts

1992 ◽  
Vol 3 (4) ◽  
pp. 319-341 ◽  
Author(s):  
S. P. Hastings ◽  
L. A. Peletier
Keyword(s):  
Asymptotic Behaviour ◽  
Physical Parameter ◽  
Dimensional Analysis ◽  
Existence And Uniqueness ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Eigenvalue Parameter ◽  
Self Similar ◽  
Similar Solutions

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

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