A semilinear heat equation with a localized nonlinear source and non-continuous initial data

2011 ◽  
Vol 34 (15) ◽  
pp. 1910-1919 ◽  
Author(s):  
Lucas C. F. Ferreira ◽  
Elder J. Villamizar-Roa
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Nonlinear Source ◽  
Semilinear Heat Equation

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.

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.

Classification of Global and Blow-up Sign-changing Solutions of a Semilinear Heat Equation in the Subcritical Fujita Range : Second-order Diffusion

2014 ◽  
Vol 14 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Enzo Mitidieri ◽  
Stanislav I. Pohozaev
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Blow Up ◽  
Second Order ◽  
Seminal Paper ◽  
Semilinear Heat Equation ◽  
Sign Changing Solutions

AbstractIt is well known from the seminal paper by Fujita [22] for 1 < p < puwith arbitrary initial data u

Global Sign-changing Solutions of a Higher Order Semilinear Heat Equation in the Subcritical Fujita Range

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Victor A. Galaktionov ◽  
Enzo Mitidieri ◽  
Stanislav I. Pohozaev
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Initial Function ◽  
Blow Up ◽  
Higher Order ◽  
Semilinear Heat Equation ◽  
Sign Changing Solutions

AbstractA detailed study of two classes of oscillatory global (and blow-up) solutions was began in [20] for the semilinear heat equation in the subcritical Fujita rangewith bounded integrable initial data u(x, 0) = uwith the same initial data u∫ ui.e., as for (0.1), any such arbitrarily small initial function u

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