### 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 |

### Keywords

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

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## Cite this

*Rocky Mountain Journal of Mathematics*,

*28*(4), 1279-1301.