半线性热传导方程Cauchy问题全局经典解的存在性与唯一性

The Existence and Uniqueness of the Global Classic Solution to the Cauchy Problem of Semilinear Heat Transfer Equations

本文证明了热传导方程Cauchy问题当实数α>3时,只要初值φ(x)在某些Sobolev空间中的范数充分小,就有唯一的全局经典解,且当t→+∞时,这个解具有一定的衰减性。本文所用的方法使得Cauchy问题中的α的值同解与初值所在的空间紧密联系,α的值越大,解的性质越好。

we investigated the existence and uniqueness of the cauchy problem of semilinear heat transfer equation. We proved that for α>3, if the initial value ψ(x) is sufficiently small in some Sobolev spaces, there exists a unique global classic solution for the Cauchy problem. And the solution decays as t→+ ∞. The method used in this paper makes the value of α be closely combined with the spaces in which the sloution and initial value function are defined. The larger the value of α is, the better the properties of the solution are.