We present some facts about the smoothness of solutions of the initial-boundary-value problems for the parabolic system of partial differential equations

$$ {\displaystyle \begin{array}{c}{u}_t-{\left(-1\right)}^mP\left(x,t,{D}_x\right)u=f\left(x,t\right)\kern1em \mathrm{in}\kern1em \Omega \times \left(0,T\right),\\ {}\frac{\partial^ju}{\partial {v}^j}=0\kern1em \mathrm{on}\kern1em \left(\mathrm{\partial \Omega}\backslash M\right)\times \left(0,T\right),\\ {}u\left(x,0\right)=0,\end{array}} $$

in a domain of dihedral type Ω_T , where P is an elliptic operator with variable coefficients. It is shown that the regularity of solutions depends on the distribution of eigenvalues of the corresponding spectral problems. The obtained results can be useful for understanding the asymptotics of weak solutions near the singular edges of dihedral domains.

中文翻译：

---

二面体域抛物系统的初边值问题

我们提出了一些偏微分方程的抛物线型方程组初边值问题解的光滑性的事实。

$$ {\ displaystyle \ begin {array} {c} {u} _t-{\ left（-1 \ right）} ^ mP \ left（x，t，{D} _x \ right）u = f \ left（ x，t \ right）\ kern1em \ mathrm {in} \ kern1em \ Omega \ times \ left（0，T \ right），\\ {} \ frac {\ partial ^ ju} {\ partial {v} ^ j} = 0 \ kern1em \ mathrm {on} \ kern1em \ left（\ mathrm {\ partial \ Omega} \反斜杠M \ right）\ times \ left（0，T \ right），\\ {} u \ left（x， 0 \ right）= 0，\ end {array}} $$

在双面型Ω的结构域_Ť，其中P是一个椭圆算变系数。结果表明，解的正则性取决于相应频谱问题特征值的分布。获得的结果对于理解二面体域奇异边缘附近的弱解的渐近性可能是有用的。