Degenate Parabolic Boundary Value Problems for Quasilinear ion Equations in L^1(0,1)

• Main Authors: Suen Shin Sue, 蘇孫鑫
• Other Authors: Chin-Yuan Lin
• Format: Others
• Language: en_US
• Published: 1993
• Online Access: http://ndltd.ncl.edu.tw/handle/84652943766114984836

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.