Qualitative Properties Of A Kind Of Quasilinear Parabolic Equation

In this thesis, taking a(u) = 1/mum as a model, the continuous dependency on the initial date and on nonlinear properties of parabolic equation ut = (a(u))xx are considered. The existence and the life span of solutions to some mixed initial-boundary value problems are also investigated.For the Cauchy problem, it is proved that there is no solution in L1 as m - 1, however, there exists a unique global solution u(x,t, m) as m > - 1, such that(1) lim R |u(x,t,m) - u(x, t,m0)\dx = 0, t 0;(2) lim |u(x, t, m) - u(x, t, m0)| - 0 uniformly with respect to |x| lfor any given l, t 0;(3) lim |u(x,t,m) - u(x,t, mo)| = 0 uniformly with respect to (x,t) [-l, l] x [ , T] for any given l > 0,0 < < T and -1 < m0 < 1;(4) 0T R |u(x,t,m)-u(x,t,m0)|2dxdt C*|m-m0|, in which m, m0 > 0 and C* is a positive constant;(5) Ru(x,t,m)dx = Ru0(x)dx, t 0;(6) R u(x,t,m) - u(x,t,m)\dx R |U0(X) - 0(x)\dx, t 0.For the mixed initial- boundary value problem, it is proved that there exists a unique global solution u(x, t, m) to the problem with homogeneous Neumann boundary conditions for m 6 R and lim |u(x,t,m) - u(x,t,m0)| = 0uniformly with respect to x [0,1] for any given t > 0 in which m0 R, moreover, some properties obtained for the Cauchy problem are also valid. For another kind of problem with nonlinear boundary conditions, we get not only the existence and uniqueness, but also the life-span of solution.If m < 1 and u0 0, in order to overcome difficulties coming from the occurrence of essential singularities near t = 0 we use different methods fordifferent problems.