Singular Integral Operators and Nonlinear Problems

我们的结果可以概括如下1.首先，我们从泛函分析的角度研究了概率论中带边界条件的马尔可夫过程的构造问题。我们的方法的特点是广泛使用了奇异积分算子理论中最新发展的思想和技术。我们构造了一个Feller半群，对应于这样一种扩散现象，即马尔可夫粒子在状态空间中跳跃和连续运动，直到它到达粒子被确定吸收的集合时死亡。2.研究了具有不定权重的扩散Logistic方程正解的存在唯一性问题，该方程模型是强空间异质性环境中的种群动力学模型。我们证明，如果有一个相对较大的有利区域(具有良好的资源，没有拥挤效应)位于距离环境边界一定距离的地方，就会出现最有利的情况。此外，我们还讨论了正定态的稳定性。第三，我们研究了燃烧理论中的半线性椭圆型边值问题，这些边值问题服从弹性Arrhenius速率定律和牛顿冷却。证明了在无因次放热速率的某些临界值下，稳定的稳态温度分布中存在着火和熄火现象。

Our results may be summarized as follows1.First, we studied from the viewpoint of functional analysis the problem of construction of Markov processes with boundary conditions in probability theory. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of singulars integral operators. We constructed a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the set where the particle is definitely absorbed2.Secondly, we studied existence and uniqueness problems of positive solutions of diffusive logistic equations with indefinite weights which model population dynamics in environments with strong spatial heterogeneity. We proved drat the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. Moreover, we discussed the stability properties for positive steady states3.Thirdly, we studied semilinear elliptic boundary value problems arising in combustion theory which obey the supple Arrhenius rate law and Newtonian cooling. We proved that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless heat evolution rate