一类非稳态热耦合方程组解的存在性

研究了一类椭圆抛物耦舍方程组解的存在性。在假设耦合系数σ（s）、k（s）∈W^1,m（R）,b∈L^∞（Ω）]^2,c∈L^∞（Ω）且满足c-1/2△↓·b≥-（k1-α）λ1条件下，λ1这-△的第一特征值。α＞0.运用Faedo—Galerkin方法构造近似解，首先得出近似解在局部时间内存在，然后得出一些近似解的先验估计证明解可以延拓到区间0，T]，利用紧性定理得出解关于时间t和n无关。最后对逼近方程取极限，得出解整体存在。若耦合系数σ（s）、k（s）退化，构造其截断函数，仍可得出解存在且有界。

Existence of Solutions to an Unsteady Thermally Coupled System

In this paper, a global solution to the coupled elliptic-parabolic system modeling a class of engineering problem with thermal effect is studied. Under the assumptions that σ（s）、k（s）∈W^1,m（R）,b∈L^∞（Ω）]^2,c∈L^∞（Ω）and c-1/2△↓·b≥-（k1-α）λ1（ λ1 denotes the first eigenvalue of - △ in Ω,α 〉 0）, we apply Faedo-Galerkin method to construct an approximate solution to the problem that exists in a local time, then prove some a priori estimates to show that the approximate solution can be extended to the interval 0, T]. By using the compact theorem, we get the global solution as n→∞. If the coupled functions σ（s） ,k（s） are degenerate,we can also get the existence of solution to the considered problem by constructing their truncation functions.