论文标题
与罗宾边界条件的线性和半分数扩散方程的比较原理
Comparison principles for the linear and semiliniar time-fractional diffusion equations with the Robin boundary condition
论文作者
论文摘要
本文的主要目的是分析与二阶均匀椭圆形的空间差分运算符以及分数SOBOLOLEV空间中的CAPUTO型分数衍生物作用的线性和半线性时间散射扩散方程。边界条件是以均匀的诺伊曼或罗宾条件的形式制定的。首先，我们处理这些初始有限价值问题的独特性和解决方案的存在。然后，我们显示解决方案的阳性属性，并得出相应的比较原理。在半线性时段扩散方程的情况下，我们还通过上和下溶液应用单调性方法。作为我们结果的应用，我们为半线性时间裂纹扩散方程的解决方案提供了一些先验估计。
The main objective of this paper is analysis of the initial-boundary value problems for the linear and semilinear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we deal with the uniqueness and existence of the solutions to these initial-boundary value problems. Then we show a positivity property for the solutions and derive the corresponding comparison principles. In the case of the semilinear time-fractional diffusion equation, we also apply the monotonicity method by upper and lower solutions. As an application of our results, we present some a priori estimates for solutions to the semilinear time-fractional diffusion equations.
