Systems Describing Electrothermal Effects with p(x)-Laplacian-like Structure for Discontinuous Variable Exponents
Systems Describing Electrothermal Effects with p(x)-Laplacian-like Structure for Discontinuous Variable Exponents
复制标题
用 p(x)-类拉普拉斯结构描述不连续变指数电热效应的系统
DOI：
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发表时间：
2016
影响因子：
2
通讯作者：
M. Liero
中科院分区：
文献类型：
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作者：
M. Bulíček;A. Glitzky;M. Liero
我们考虑一个耦合的两个椭圆型偏微分方程组，其中第一个方程中的椭圆项具有不连续指数的$p（x）$-Laplacian的性质，而在第二个方程中我们必须处理右手边的先验$L^1$项.这样的系统适合于描述各种欧姆效应，特别是那些非欧姆行为可以相对于空间变量显著变化的系统。我们证明了弱解的存在性非常弱的假设下的数据，也根据一般的结构假设模型的本构方程。主要的困难在于我们必须同时克服两个障碍-不连续变量指数和热方程的右手边。我们的存在性证明基于Galerkin近似是高度建设性的，因此似乎也适用于数值目的。
We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the $p(x)$-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori $L^1$ term on the right-hand side. Such systems are suitable for the description of various electrothermal effects, in particular, those where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to simultaneously overcome two obstacles---the discontinuous variable exponent and the $L^1$ right-hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.