In this paper we consider the Cauchy problem for the pseudo-parabolic equation:$$\begin{aligned} \dfrac{\partial }{\partial t} \left( u + \mu (-\Delta )^{s_1} u\right) + (-\Delta )^{s_2} u = f(u),\quad x \in \Omega ,~ t>0. \end{aligned}$$Here, the orders \(s_1, s_2\) satisfy \(0<s_1 \ne s_2 <1\) (order of diffusion-type terms). We establish the local well-posedness of the solutions to the Cauchy problem when the source f is globally Lipschitz. In the case when the source term f satisfies a locally Lipschitz condition, the existence in large time, blow-up in finite time and continuous dependence on the initial data of the solutions are given.

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非线性伪抛物方程的适定性

在本文中，我们考虑了伪抛物方程的柯西问题：$$ \ begin {aligned} \ dfrac {\ partial} {\ partial t} \ left（u + \ mu（-\ Delta）^ {s_1} u \ right）+（-\ Delta）^ {s_2} u = f（u），\ quad x \ in \ Omega，〜t> 0。\ end {aligned} $$在这里，顺序\（s_1，s_2 \）满足\（0 <s_1 \ ne s_2 <1 \）（扩散类型项的顺序）。当来源f为全局Lipschitz时，我们确定了柯西问题的解的局部适定性。在源项f满足局部Lipschitz条件的情况下，给出了长时间存在，有限时间内的爆发以及对解的初始数据的连续依赖的情况。