THE FIRST BOUNDARY VALUE PROBLEM FORSOLUTIONS OF DEGENERATE QUASUJNEARPARABOLIC EQUATIONS

In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation$$[left{ {begin{array}{*{20}{c}}{frac{{partial u}}{{partial t}} = sum {frac{partial }{{partial {x_i}}}left( {v(u){A_{ij}}(x,t,u)frac{{partial u}}{{partial {x_j}}}} right) + sum {{B_i}(x,t,u)} frac{{partial u}}{{partial {x_i}}}} + C(x,t,u)ubegin{array}{*{20}{c}}{}&{(x,t) in [0,T]}end{array},}{u{|_{t = 0}} = {u_0}(x),x in Omega ,}{u{|_{x in partial Omega }} = psi (s,t),0 le t le T}end{array}} right.]$$$$[left( {frac{1}{Lambda }{{left| alpha right|}^2} le sum {{A_{ij}}{alpha _i}{alpha _j}} le Lambda {{left| alpha right|}^2},forall a in {R^n},0 < Lambda < infty ,v(u) > 0begin{array}{*{20}{c}}{and}&{v(u) to 0begin{array}{*{20}{c}}{as}&{u to 0}end{array}}end{array}} right)]$$under some very weak restrictions, i.e. $[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),sum {frac{{partial {A_{ij}}}}{{partial {x_j}}}} ,sum {frac{{partial {B_i}}}{{partial {x_i}}} in overline Omega } times [0,T] times R,left| {{B_i}} right| le Lambda ,left| C right| le Lambda ,],[left| {sum {frac{{partial {B_i}}}{{partial {x_i}}}} } right| le Lambda ,partial Omega in {C^2},v(r) in C[0,infty ).v(0) = 0,1 le frac{{rv(r)}}{{int_0^r {v(s)ds} }} le m,{u_0}(x) in {C^2}(overline Omega ),psi (s,t) in {C^beta }(partial Omega times [0,T]),0 < beta < 1],[{u_0}(s) = psi (s,0).]$