Some Problems Of Nonlinear Partial Differential Equations

In this paper, we study two problems of nonlinear partial differential equations.In Chapter One, the author briefly reviews the history of viscosity solutions and the theory of blow-up and then sets forth the problems discussed in this paper.In Chapter Two, the following problem is considered:Using the theory of viscosity solutions, the following theorem is proved.Theorem: Suppose that f ( t ,x ) is continuous uniformly in x , F is continuous, andψsatisfies some smooth and growth conditions, then(I)has a unique solution u ( t ,x ) that grows at most linearly and u is continuous uniformly in x.In Chapter Three, the author considers the following problem:The author proves the following theorem:Theorem: Assume thatΩis a symmetrical and bounded domain in R^N, q = p> 2, b = 1, a < 0, a is small enough, initial value u₀=λψ, andψis nonnegative and nonzero , If u is a nonnegative radial solution of (II), then there exists aΛ₀ =Λ₀(p , a ,Ω,ψ) > 0 such that, for allλ≥Λ₀, blow-up occurs or...