So far, although there have been several articles on partial differential equations with piecewise constant arguments, as far as we know, there is no article on neither a heat equation with piecewise constant mixed arguments that includes three extra diffusion terms, delayed arguments \([t-1], [t]\) and an advanced argument \([t+1],\) or exploring qualitative properties of the equation. With the motivation to investigate elaborate and well-established qualitative properties of such an equation, in this paper, we deal with a problem involving a heat equation with piecewise constant mixed arguments and initial, boundary conditions. By using the separation of variables method, we obtain the formal solution of this problem. Because of the piecewise constant arguments, we get a differential equation and then a difference equation. With the help of qualitative properties of the solutions of the differential equation and with the behavior of the solutions of the difference equation, we investigate the existence of solutions and qualitative properties of the solutions of the problem such as the convergence of the solutions to zero, the unboundedness of the solutions and oscillations of them. In addition, two examples are given to illustrate the application of the results in particular cases.

到目前为止，尽管有几篇关于具有分段常数参数的偏微分方程的文章，但据我们所知，还没有一篇关于具有分段常数混合参数的热方程的文章，其中包括三个额外的扩散项，延迟参数\（[ t-1]，[t] \）和高级参数\（[t + 1]，\）或探索方程的定性性质。为了研究这种方程式的精细和完善的定性性质，在本文中，我们处理了一个涉及具有分段常数混合参数和初始边界条件的热方程式的问题。通过使用变量分离方法，我们可以得到此问题的形式化解决方案。由于分段常数参数，我们得到了一个微分方程，然后是一个差分方程。借助微分方程解的定性性质和差分方程解的性质，我们研究了问题的解的存在性和定理性质，例如零解的收敛性，解的无限性和它们的振荡。另外，给出了两个例子来说明结果在特定情况下的应用。