| Summary: | 碩士 === 國立中央大學 === 數學系 === 81 === This thesis is motivated by the paper [8], which studies the
following degenerate parabolic problem u.sub.(t)(x,t)=.prtl./.
prtl.x(.prtl./.prtl.x(.beta.u)) -.prtl./.prtl.x(.alpha.u),(x,t).
in.(0,1)*(0,.inf.) (1) .beta.(u(1,0))=0, .prtl./.prtl.x(.beta.(
u(0,t)))-.alpha .(u(0,t))-.gamma.sub.(0)(.beta.(u(0,t)))=0 u(
x,0)=u.sub.(0)(x) Our thesis here is to consider u.sub.(t)(x,
t)=.prtl./.prtl.x.phi.(x,.prtl./.prtl.x( .beta.u))-.prtl./.prtl.
x(.alpha.u),(x,t).in.(0,1)*(0,.inf. (2) .phi.(0,.prtl./.prtl.x(.
beta.(u(0,t)))-.alpha.(u(0,t)) u.sub.(t)(x,t)=.prtl./.prtl.x.
phi.(x,u.sub.(x)) -.gamma.sub.(0)(.beta.(u(0,t)))=0,.beta.(u(1,
t))=0 +f(x,u),(x,t).in.(0,1)*(0,.inf.) u(x,0)=u.sub.(0)(x)
(.phi.(0,u.sub.(x)(0,t)),-.phi.(1,u.sub.(x)(1,t))).in. Thus (1)
is a special case of (1) if we take .phi.(x,.xi)=.xi We shall
solve (2) by using nonlinear operator semigroup theory [1,3,5,6
].This approach is to write (2) as an ordinary differential
equation: du/dt=Au, t>0 (3) u(0)=u.sub.(0) in the real Banach
space L.sup.(1)(0,1), where the boundary condition is absorbed
in the definition of the domain D(A) of the nonlinear operator
A. A is shown to be essentially m-dissipative, and then by the
Crandall-Liggett theorem [4,5] the nonlinear operator T(t)
exists.T(t).u.sub(0) will be a unique generalized solution to
(3)[2],and then (2) is,in turn solved.
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