强和弱外推的$L^2$ --具有常数时滞的Heat Equation的解
Strong and Mild Extrapolated $L^2$-Solutions to the Heat Equation with Constant Delay
摘要：
在有界区域上，提出了一类具高阶导数时滞的非齐次热方程的希尔伯特空间解的理论，该方程具有非齐次Dirichlet边界条件.在相当弱的正则性假设下，我们证明了一个适定性结果，并给出了一个明确的表示的解决方案。此外，我们证明了指数衰减率的能量耗散的情况下。我们还表明，低阶正则化导致不适定性，也为高阶方程。最后，给出了一个具有物理相关常数的应用。
英文摘要：
We propose a Hilbert space solution theory for a nonhomogeneous heat equation with delay in the highest order derivatives with nonhomogeneous Dirichlet boundary conditions in a bounded domain. Under rather weak regularity assumptions on the data, we prove a well-posedness result and give an explicit representation of solutions. Further, we prove an exponential decay rate for the energy in the dissipative case. We also show that lower order regularizations lead to ill-posedness, also for higher-order equations. Finally, an application with physically relevant constants is given.
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