We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the decay rates of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffusion) in a limit procedure when we rescale the nonlocal kernel in a suitable way.

我们证明了演化方程的存在性，唯一性和几个定性性质，它们结合了作用在不同子域中的局部和非局部扩散算子，并以这样的方式耦合，使得所得的演化方程是能量函数的梯度流。在后两种情况下，我们用零边界数据处理柯西，诺伊曼和狄利克雷问题。对于前两个问题，我们证明该模型保留了总质量。我们还大量研究了溶液的衰减率。最后，我们证明了当我们以适当的方式重新缩放非局部内核时，可以在极限过程中恢复通常的热方程（局部扩散）。