The Qualitative Properties Of A Nonlinear Degenerate Parabolic Equation With Periodic Source

This thesis discusses two problems. The ?rst one is the Cauchy problem of a nonlinear degenerate parabolic equation with periodic source and the second one is the geometric properties of the Cauchy problem of a nonlinear degenerate parabolic equation with periodic source.1. Consider the Cauchy problem of a nonlinear degenerate parabolic equation with periodic source: ut= ?um+ θupsin t(p > m > 1). Conclusions:(1)The existence, uniqueness, boundedness and stability of the solution;(2)The estimate of gradient;(3)The L1 estimate of solutin;(4)The estimate for Laplace lower bound of the pressure term;(5)The estimate about the time derivative: ut-[α1+ α2(eα0t- 1)-1]u;(6)The asymptotic behavior of the solution(the spread of the solution): sup x∈Hu(t)|x| [f(t, θ)]μ.2. Consider the geometric properties of the Cauchy problem of a nonlinear degenerate parabolic equation with periodic source: ut= ?um+ upsin t(p > m > 1). Conclusions:(1) For any t ∈ R+, δ > 1, the surface S(t) is the complete Riemannian manifold in space RN +1, and it is tangent to the space RNat lower-dimension mani?od ?H0(t); The surface u = u(x, t) is tangent to the hyperplane W(t) at ?Hu(t), the maximum value of u(x, t). Where,?H0(t) ={x x ∈ RN, u(x, t) = 0},?Hu(t) ={x x ∈ RN, u(x, t) =[M1-p-(p- 1)(1- cos t)]-11-p}.(2)The solution’s curvature change about space variable at the maximum points:k = uxx(1 + u2x)32 C1(eC0t- 1)-1+ C2.Here, k is the curvature of solution u = u(x, t) at the maximum points for any t > 0.