Singular Integral Operators and Nonlinear Problems

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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