论文标题
时间裂缝多孔培养基以及更通用的非线性和退化进化界面方程的有界弱解
Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations
论文作者
论文摘要
我们证明存在一个有界弱解，可以通过有限的可测量系数明确依赖于时间，而对退化的准线性次扩散问题存在。所涉及的局部差异操作员W.R.T.中的内核时间属于$ {\ cal pc} $内核的大类。特别是，包括小于1的顺序的分数时间导数的情况。证明中的一个关键成分是Aubin-Lions类型的新紧凑性标准，该标准涉及按及时根据Integro-Difterention Operator定义的功能空间。解决方案的界限是通过De Giorgi迭代技术获得的。通过$ L_1 $ - 征收估计，足够的常规解决方案被证明是唯一的。
We prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w.r.t. time belongs to the large class of ${\cal PC}$ kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the De Giorgi iteration technique. Sufficiently regular solutions are shown to be unique by means of an $L_1$-contraction estimate.
