【摘要】 本文将非线性半群理论用于二阶拟线性退缩抛物方程的Cauchy问题(1),(2),证明了广义解的存在性。更多还原

【Abstract】 In this paper, the theory of nonlinear semigroups is applied to the Cauchy problem for the quasilinear degenerate parabolic equation （1） of second order.We first define an operato A₀ as follows:Let u（x）∈L¹（R）∩BV（R）, w（x）∈L₁（R）. Then we say that the operator A₀ is defined at u, u∈D（A₀） （the domain of A₀ ）, and w∈A₀u, if:1) There exists a function s（x）∈_loc²（R）, such thatHere the notation denotes the mean value of composition of p（u） and u（x）.Further we define A as, the closure of A₀.It is essential to prove that the operator A is accretive and R（I+λA）=L₁（R）（λ>0）.Once these are proved, applying the socalled Generation Theorem, we can conclude that the operator generates a nonlinear contractive semigroup S（t） on L¹（R）. Thus, for any u₀（x）∈L¹（R）, we get a "generalized" solution S（t）u₀（x） of the Cauchy problem （1）, （2）.If the initial value u₀（x） is much smoother and some restriction on the behaviour at infinity is assumed, then more informations about the solution of the Cauchy problem （1）, （2） may be obtained.更多还原