We consider the heat equation defined by a generalized measure theoretic Laplacian on [0, 1]. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure \(\mu \), where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding heat equations. This provides, for example, an interpretation for the mathematical model of heat diffusion on a bar with gaps in that the solution to the corresponding heat equation behaves approximately like the heat flow on a bar with sufficiently small mass on these gaps.

我们考虑由[0，1]上的广义测度理论Laplacian定义的热方程。该方程式描述了棒中的热扩散，因此棒的质量分布由非原子的Borel概率度量\（\ mu \）给出，这里我们不假设存在严格的正质量密度。我们表明，弱测度收敛意味着在强分解意义上相应的广义拉普拉斯算子的收敛。我们证明遵循均匀范数的强半群收敛，这意味着相应热方程解的均匀收敛。例如，这提供了对具有间隙的棒上的热扩散的数学模型的解释，其中对相应热方程的解的行为近似类似于在这些间隙上质量足够小的棒上的热流。