We study for nonlinear Kirchhoff's model of pseudo parabolic type by considering its two different problems.$ \bullet $ For initial value problem, we obtain the results on the existence and regularity of solutions. Moreover, we also prove that the solutions $ u $ corresponding with $ \beta < 1 $ of the problem convergence to $ u $ for $ \beta = 1 $.$ \bullet $ For final value problem, we show that the ill-posed property in the sense of Hadamard is occurring. Using the Fourier truncation method to regularize the problem. We establish some stability estimates in the $ H^1 $ and $ L^p $ norms under some a-priori conditions on the sought solution.

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关于基尔霍夫型分数阶非经典扩散方程的初终值问题

我们通过考虑它的两个不同问题来研究伪抛物型非线性基尔霍夫模型。$ ​​\bullet $ 对于初值问题，我们得到了解的存在性和正则性的结果。此外，我们还证明了与问题的 $ \beta < 1 $ 对应的解决方案 $ u $ 收敛到 $ u $ for $ \beta = 1 $.$ \bullet $ 对于终值问题，我们证明了Hadamard 意义上的构成财产正在发生。使用傅里叶截断方法对问题进行正则化。我们在寻求解决方案的某些先验条件下，在 $ H^1 $ 和 $ L^p $ 范数中建立了一些稳定性估计。