### Abstract

We consider the Cauchy problem for a two-dimensional semi-linear heat equation with radial symmetry u_{t} := u_{rr}+u_{r}/r + e^{u} in R^{+} × (0, T); with smooth, bounded initial data u_{0}(r). We prove that the solution u(r, t) becomes strictly monotone in time, u_{t} > 0, at any point where u is large enough. The proof is based on intersection comparison of u(r, t) with the set {w(·)} of stationary solutions satisfying w? + w' /r + e^{w} = 0 for r > 0. The above monotonicity result is shown to depend essentially on the global structure of the set {w}. The same result is found to hold for positive solutions u to the equation with power nonlinearity u_{t} = ?u + u^{p}, 1 < p < (N + 2)/(N - 2)+. Several generalizations to boundary value problems and quasi-linear equations are given.

Original language | English |
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Pages (from-to) | 1279-1301 |

Number of pages | 23 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 28 |

Issue number | 4 |

Publication status | Published - Dec 1998 |

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### Keywords

- Intersection comparison
- Maximum principle
- Semilinear heat equation
- Stationary solutions

### Cite this

*Rocky Mountain Journal of Mathematics*,

*28*(4), 1279-1301.

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*Rocky Mountain Journal of Mathematics*, vol. 28, no. 4, pp. 1279-1301.

**Monotonicity in time of large solutions to a nonlinear heat equation.** / Galaktionov, V. A.; Lacey, A. A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Monotonicity in time of large solutions to a nonlinear heat equation

AU - Galaktionov, V. A.

AU - Lacey, A. A.

PY - 1998/12

Y1 - 1998/12

N2 - We consider the Cauchy problem for a two-dimensional semi-linear heat equation with radial symmetry ut := urr+ur/r + eu in R+ × (0, T); with smooth, bounded initial data u0(r). We prove that the solution u(r, t) becomes strictly monotone in time, ut > 0, at any point where u is large enough. The proof is based on intersection comparison of u(r, t) with the set {w(·)} of stationary solutions satisfying w? + w' /r + ew = 0 for r > 0. The above monotonicity result is shown to depend essentially on the global structure of the set {w}. The same result is found to hold for positive solutions u to the equation with power nonlinearity ut = ?u + up, 1 < p < (N + 2)/(N - 2)+. Several generalizations to boundary value problems and quasi-linear equations are given.

AB - We consider the Cauchy problem for a two-dimensional semi-linear heat equation with radial symmetry ut := urr+ur/r + eu in R+ × (0, T); with smooth, bounded initial data u0(r). We prove that the solution u(r, t) becomes strictly monotone in time, ut > 0, at any point where u is large enough. The proof is based on intersection comparison of u(r, t) with the set {w(·)} of stationary solutions satisfying w? + w' /r + ew = 0 for r > 0. The above monotonicity result is shown to depend essentially on the global structure of the set {w}. The same result is found to hold for positive solutions u to the equation with power nonlinearity ut = ?u + up, 1 < p < (N + 2)/(N - 2)+. Several generalizations to boundary value problems and quasi-linear equations are given.

KW - Intersection comparison

KW - Maximum principle

KW - Semilinear heat equation

KW - Stationary solutions

UR - http://www.scopus.com/inward/record.url?scp=0032223466&partnerID=8YFLogxK

M3 - Article

VL - 28

SP - 1279

EP - 1301

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

SN - 0035-7596

IS - 4

ER -