We consider the heat equation coupled with the Maxwell system, the Ampère-Maxwell equation being coupled to the heat equation by the permittivity, which depends on the temperature due to thermal agitation, and the heat equation being coupled to the Maxwell system by the volumic heat source term. Our purpose is to establish the existence of a local-in-time solution to this coupled problem. Firstly, fixing the temperature distribution, we study the resulting Maxwell system, a nonautonomous system due to the dependence of the permittivity on the temperature and consequently on time, by using the theory of evolution systems. Next, we return to our coupled problem, introducing a fixed-point problem in the closed convex set $K(0;R):=\{z\in \overline{B}(0;R);z(0)=0\}$ of the Banach space $C^1([0,T];C^1(\overline{\mathrm{\Omega }}))$ and proving that the hypotheses of Schauder's theorem are verified for R sufficiently large. The construction of the fixed-point problem is nontrivial as we need $K(0;R)$ to be stable.

我们考虑与 Maxwell 系统耦合的热方程，Ampère-Maxwell 方程通过介电常数与热方程耦合，介电常数取决于热搅动引起的温度，而热方程通过体积热与 Maxwell 系统耦合源术语。我们的目的是建立这个耦合问题的本地时间解决方案的存在。首先，固定温度分布，我们使用演化系统理论研究由此产生的麦克斯韦系统，这是一个非自治系统，因为介电常数与温度和时间相关。接下来，我们回到我们的耦合问题，在闭凸集中引入一个不动点问题$钾(0;\mathrm{电阻})：=\{z\in \overline{乙}(0;\mathrm{电阻});z(0)=0\}$ 巴拿赫空间 $C^1([0,吨];C^1(\overline{\mathrm{\Omega }}))$并证明 Schauder 定理的假设在R足够大时得到验证。根据我们的需要，不动点问题的构建是非常重要的$钾(0;\mathrm{电阻})$ 要稳定。