【作者】 王华；

【导师】 李胜家；

【作者基本信息】 山西大学 ， 基础数学， 2015， 博士

【摘要】 本文的研究内容主要有三个,即：半线性变指数方程解的爆破；非柱面区域上波动方程的精确能控性和关于F-无穷Laplace算子的方程的黏性解.首先研究了一类半线性抛物和双曲方程的爆破问题,然后利用HUM研究了一维非柱面区域上波动方程的精确能控性问题,最后利用Perron方法研究了关于F-无穷Laplace算子的Dirichlet边值问题黏性解的存在唯一性.本文分为五章.第一章是引言,主要介绍本文的研究背景,国内外研究现状及本文的一些主要结果.第二章主要研究变指数半线性发展方程解的爆破：本章主要讨论了如下抛物方程(其中u0(x)≥0)：其中Ω(?)Rn(n>3)有界且具有Lipschitz连续边界aQ；以及双曲方程：其中u0(x),u1(x)≥0且它们都不恒为零Ω(?)Rn(n≥3)有界且具有Lipschitz连续边界aQ.如果具有一定的初始条件以及变指数函数p(x)满足假设条件：(2)对所有满足条件|z-ξ|<1的z,ξ∈Ω,有|p(z)-p(ζ)I≤ω(|z-ζ|)成立,其中ω满足通过构造一控制函数以及利用Sobolev嵌入定理得到这两个方程的解对于小的正(初始)能量在有限时刻是爆破的.第三章研究论文的第二个主要内容：一维波动方程在非柱面区域上的精确能控性.本章中,对于满足条件α(0)=1,α’单调以及0<c1≤α’(t)≤c2<1(c1,c2是常数)的二次连续可微函数α:[0,∞)→(0,∞),我们研究了非柱面区域QTα={(y,t)∈R2|0<y<α(t),t∈(0,T)}上的一维波动方程：和其中控制v∈L2(0,T).通过应用HUM,我们得到这个方程的Dirichlet边界控制在一个端点处的精确能控性结果.同时我们给出仅依赖于c1和c2的可控时刻的估计.第四章研究论文的第三个主要内容：关于F-无穷Laplace算子的Dirichlet边值问题黏性解的存在唯一性.本章中,对于Rn\{0}中满足条件Hess(F2)正定的正的一阶正齐次C2函数F,我们定义了F-无穷Laplace算子△F：∞和正规化的F-无穷Laplace算子△F,∞N.对于Rn中的有界区域Ω,Ω上满足条的连续函数f,以及g∈C((?)Ω),利用Perron方法得到关于F-无穷Laplace算子的Dirichlet边值问题和正规化的F-无穷Laplace算子的Dirichlet边值问题的黏性解的存在唯一性结论.此外我们也用Perron方法得到了齐次F-无穷Laplace方程的Dirichlet边值问题的黏性解的存在性.第五章是总结与展望.概括了论文的主要结果,并对进一步的研究工作做了展望.更多还原

【Abstract】 The main contents of this thesis consist of three parts. That is blow-up of solutions for some semilinear equations with variable exponent; an exact controllability problem of a wave equation in non-cylindrical domains and vis-cosity solutions to equations involving F-infinity Laplacian. Firstly we study blow-up of solutions for a semilinear parabolic equation and hyperbolic equa-tion; Secondly we study an exact controllability problem of a wave equation in non-cylindrical domains by using the Hilbert uniqueness method; Lastly we study existence and uniqueness of viscosity solutions to the Dirichlet boundary value problem involving F-infinity Laplacian by applying Perron’s method.This paper consists of five chapters.In chapter 1, some research background, the research advances of some related works are given. Moreover, we list the main results obtained in this thesis.Chapter 2 is devoted to the study on the blow-up of solutions for some semilinear evolution equations with variable exponent:In this chapter, we consider the parabolic equation (where u0(x)≥0): where Ω(?)Rn(n≥3) be a bounded domain with Lipschitz continuous boundary (?)Ω; and the hyperbolic equation: where u0(x),u1(x)≥0 and they are not identically zero,Ω(?)Rn(n≥3)be a bounded domain with Lipschitz continuous boundary aQ.With certain initial data and variable exponent p(x)satisfying the condi-tions:(1)1<p-:=infx∈Ωp(x)≤p(x)≤p+:=supx∈Ωp(x)≤n+2/n-2;(2)|p(z)-p(ζ)|≤ω(z-ζ|), for all z,ζ∈Ω with |z-ζ|<1,where ω satisfies we prove that the solutions of these two equations blow up in finite time for small positive(initial)energy.We do this by constructing a control function and applying the suitable embedding theorems.Chapter 3 is devoted to the study of the second main content:an exact controllability of a wave equation in non-cylindrical domains.In this chapter,for a twice continuous differentiable function α:[0,∞)→ (0,∞)which satisfies that α(0)=1,α’ is monotone and 0<c1≤α’(t)≤ c2<1 for some constants c1,c2,In a non-cylindrical domain QTα={(y,t)∈R2|0<y<α(t),t∈(0,T)}, we study the exact controllability of a one-dimensional wave equation: and where the control v∈L2(0,T).By using the Hilbert Uniqueness Method, we obtain the exact controlla-bility results of this equation with Dirichlet boundary control on one endpoint. We also give estimates on the controllability time that depends only on c1 and c2.Chapter 4 is devoted to studying the third main content:existence and uniqueness of viscosity solutions to the Dirichlet boundary value problem in-volving F-infinity Laplacian.In this chapter, for a positively homogeneous of degree 1 function F: Rn\{0}→(0,+∞) which is of class C2 and satisfies that HessF2 is posi-tive definite, we define the F-infinity Laplacian ΔF;∞ and the normalized F-infinity Laplacian ΔF;∞N For a bounded domain Ω in Rn, f∈C(Ω) with we obtain existence and uniqueness results of viscosity solutions to the Dirichlet boundary value prob- lem involving F-infinity Laplacian and to the Dirichlet boundary value problem of normalized F-infinity Lapla-cian by using Perron’s method. We also obtain existence result of viscosity solutions to the Dirichlet boundary value problem of homogeneous F-infinity Laplacian equation by using Perron’s method.Chapter 5 is the summary and perspective. In section 1 of this chapter, we summarizes the main works of this thesis. In section 2 of this chapter, further research work are described.更多还原