Viscosity Solutions、Blow-up Of Partial Differential Equations And Related Problems

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.