The Cauchy-Dirichlet problem for the genuinely nonlinear ultra-parabolic equation with the piece-wise smooth minor term is considered. The minor term depends on a small positive parameter and collapses to the one-sided Dirac delta function as this parameter tends to zero. As the result, we arrive at the limiting initial-boundary value problem for the impulsive ultra-parabolic equation. The peculiarity is that the standard entropy solution of the problem for the impulsive equation generally is not unique. In this report, we propose a rule for selecting the ‘proper’ entropy solution, relying on the limiting procedure in the original problem incorporating the smooth minor term.

考虑具有分段平滑次项的真正非线性超抛物方程的 Cauchy-Dirichlet 问题。次要项取决于一个小的正参数，并随着该参数趋于零而折叠为单边狄拉克三角函数。结果，我们得到了脉冲超抛物线方程的极限初始边界值问题。其特点是脉冲方程问题的标准熵解通常不是唯一的。在本报告中，我们提出了一个选择“适当”熵解的规则，依赖于原始问题中包含平滑次要项的限制过程。