Cauchy Problem For Singular Semilinear Evolution Equations And The Equation Group

Since 1966 Fujita published his paper, people pay more attention to initial problem of nonlinear development equations. The development equation is from without singular to with singular and the development equations system is from without singular to with singular. After forty years several hundreds papers happen to the internation. Development equations and system with nonlinear and singular arise from various models in physics, chemistry, biology and engineering. In these years, people pay more attention to these problems. We study the Cauchy problem of nonlinear development equations and system in this paper.In chapter 1, we introduce the history and present of the nonlinear development equations. In the meantime, we show the nonlinear singular development equations and system discussed in the paper and corresponding results.In chapter 2, we study the Cauchy problem of solution for a class of semilinear singular development equations with nonzero initial values.where t>ε₀,r >1, is a continuous, bounded, nonnegative and not identically equal to zero. Methods used in this chapter come primarily from references[1][3].In chapter 3, we discuss the semilinear singular development equations system.where t >0 , x∈R^N, f(x) and g(x) are continuous, bounded, nonnegative. We show theuniqueness and blow-up of solution. Methods used in this chapter come primarily from references[2][6].