### Abstract

We construct blow-up patterns for the quasilinear heat equation u_{t} = ∇ · (k(u)∇u) + Q(u) (QHE) in Ω × (0, T), Ω being a bounded open convex set in ℝ^{N} 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)/u^{p} 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 u_{t} = ∇u + u^{p}. (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(e^{s}))ds = ∞, where f(v) is a monotone solution of the ODE f′(v) = Q(f(v))/v^{p} 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

*Nonlinear Differential Equations and Applications*,

*3*(3), 269-286. https://doi.org/10.1007/BF01194067

}

*Nonlinear Differential Equations and Applications*, vol. 3, no. 3, pp. 269-286. https://doi.org/10.1007/BF01194067

**On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation.** / Bebernes, Jerry; Bressan, Alberto; Galaktionov, Victor A.

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 -