A fully implicit finite difference scheme with second-order time evolution for strong nonlinear diffusion equation of divergence type is studied. Theoretical analysis is established under a coercive condition reflecting the diffusion characteristics of the strong nonlinear equation. Some new reasoning techniques are developed to overcome the difficulties caused by the strong nonlinearity of the divergence diffusion operator. The existence of the fully implicit finite difference solution is proved by constructing an appropriate mapping in the fixed point argument and bounding the solution as well as its first- and second-order spatial difference quotients tactfully.

中文翻译：

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强非线性扩散方程具有二阶时间演化的完全隐式解的存在性

研究了发散型强非线性扩散方程具有二阶时间演化的完全隐式有限差分格式。在强制条件下建立理论分析，以反映强非线性方程的扩散特性。开发了一些新的推理技术来克服由发散扩散算子的强非线性引起的困难。通过在不动点参数上构造适当的映射并定界界定解及其一阶和二阶空间差商，证明了完全隐式有限差分解决方案的存在。