Abstract
We construct blow-up patterns for the quasilinear heat equation ut = ∇ · (k(u)∇u) + Q(u) (QHE) in Ω × (0, T), Ω being a bounded open convex set in ℝN with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreover k(u) and Q(u)/up with a fixed p > 1 are of slow variation as u → ∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation ut = ∇u + up. (SHE) We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption ∫∞ k(f(es))ds = ∞, where f(v) is a monotone solution of the ODE f′(v) = Q(f(v))/vp defined for all v ≫ 1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 269-286 |
| Number of pages | 18 |
| Journal | Nonlinear Differential Equations and Applications |
| Volume | 3 |
| Issue number | 3 |
| DOIs | |
| State | Published - Jan 1 1996 |
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All Science Journal Classification (ASJC) codes
- Analysis
- Applied Mathematics
Cite this
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On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation. / Bebernes, Jerry; Bressan, Alberto; Galaktionov, Victor A.
In: Nonlinear Differential Equations and Applications, Vol. 3, No. 3, 01.01.1996, p. 269-286.Research output: Contribution to journal › Article
TY - JOUR
T1 - On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation
AU - Bebernes, Jerry
AU - Bressan, Alberto
AU - Galaktionov, Victor A.
PY - 1996/1/1
Y1 - 1996/1/1
N2 - We construct blow-up patterns for the quasilinear heat equation ut = ∇ · (k(u)∇u) + Q(u) (QHE) in Ω × (0, T), Ω being a bounded open convex set in ℝN with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreover k(u) and Q(u)/up with a fixed p > 1 are of slow variation as u → ∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation ut = ∇u + up. (SHE) We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption ∫∞ k(f(es))ds = ∞, where f(v) is a monotone solution of the ODE f′(v) = Q(f(v))/vp defined for all v ≫ 1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.
AB - We construct blow-up patterns for the quasilinear heat equation ut = ∇ · (k(u)∇u) + Q(u) (QHE) in Ω × (0, T), Ω being a bounded open convex set in ℝN with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreover k(u) and Q(u)/up with a fixed p > 1 are of slow variation as u → ∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation ut = ∇u + up. (SHE) We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption ∫∞ k(f(es))ds = ∞, where f(v) is a monotone solution of the ODE f′(v) = Q(f(v))/vp defined for all v ≫ 1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.
UR - http://www.scopus.com/inward/record.url?scp=0009329666&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0009329666&partnerID=8YFLogxK
U2 - 10.1007/BF01194067
DO - 10.1007/BF01194067
M3 - Article
AN - SCOPUS:0009329666
VL - 3
SP - 269
EP - 286
JO - Nonlinear Differential Equations and Applications
JF - Nonlinear Differential Equations and Applications
SN - 1021-9722
IS - 3
ER -