Recently theory of p-adic wavelets started to be actively used to study of the Cauchy problem for nonlinear pseudo-differential equations for functions depending on the real time and p-adic spatial variable. These mathematical studies were motivated by applications to problems of geophysics (fluids flows through capillary networks in porous disordered media) and the turbulence theory. In this article, using this wavelet technique in combination with the Schauder fixed point theorem, we study the solvability of nonlinear equations with mixed derivatives, p-adic (fractional) spatial and real time derivatives. Furthermore, in the linear case we find the exact solution for the Cauchy problem. Some examples are provided to illustrate the main results.

中文翻译：

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求解非线性p -Adic伪微分方程：将小波基与Schauder不动点定理组合

最近，p -adic小波理论开始被积极地用于研究函数的非线性伪微分方程的柯西问题，该函数取决于实时和p -adic空间变量。这些数学研究的动机是应用到地球物理问题（流体在多孔无序介质中通过毛细网络流动）和湍流理论。在本文中，结合小波技术和Schauder不动点定理，我们研究了带有混合导数，p- adic（分数）空间和实时导数的非线性方程的可解性。此外，在线性情况下，我们找到了柯西问题的精确解。提供了一些示例以说明主要结果。