一类具对数源的分数阶p-Laplacian方程解的存在性

Existence of Solutions for a Class of Fractional p-Laplacian Equations with Logarithmic Source Terms

摘要: 本文研究了一类具有非线性阻尼和对数源的分数阶p-Laplacian双曲方程弱解的存在性，探讨了弱解存在的适当条件。本文采用Faedo-Galerkin有限逼近方法构造近似解序列，结合不等式技巧建立一致先验估计。借助对数不等式、Hölder不等式与分数阶Sobolev嵌入定理对对数源项进行放缩。利用广义Gronwall不等式得到近似解能量泛函关于逼近参数与时间的全局一致有界性。基于Banach-Alaoglu定理提取弱、弱*收敛子列，利用Aubin-Lions紧性引理得到近似序列的强收敛性，进而推出逐点收敛，实现对对数非线性项的极限交换。同时，通过对偶空间范数估计验证二阶时间导数的一致有界性。最终在合适的初值正则性假设下，严格证明该分数阶双曲问题局部弱解的存在性。

Abstract: This paper investigates the existence of weak solutions to a fractional p-Laplacian hyperbolic equation with nonlinear damping and logarithmic source terms, and explores appropriate conditions guaranteeing the existence of weak solutions. The Faedo-Galerkin finite approximation method is adopted to construct sequences of approximate solutions, and uniform a priori estimates are established via inequality techniques. Logarithmic inequalities, the Hölder inequality and fractional Sobolev embedding theorems are utilized to estimate the logarithmic source term. The generalized Gronwall inequality yields the global uniform boundedness of the energy functional of approximate solutions with respect to the approximation parameter and time. Weak and weak* convergent subsequences are extracted by virtue of the Banach-Alaoglu theorem, and the Aubin-Lions compactness lemma is employed to derive strong convergence of the approximate sequences, which further implies pointwise convergence and enables the interchange of limits for the logarithmic nonlinear term. Meanwhile, uniform boundedness of the second-order time derivative is verified through norm estimates in the dual space. Finally, under suitable regularity assumptions on initial data, the existence of local weak solutions to this fractional hyperbolic problem is rigorously proved.