THE FIRST BOUNDARY VALUE PROBLEM FORSOLUTIONS OF DEGENERATE QUASUJNEARPARABOLIC EQUATIONS |
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Authors: | Dong Guangchang |
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Affiliation: | Department of Mathematics^ Zhejiang University, Hangzhou, Zhejiang, China. |
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Abstract: | 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. $[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),sum {frac{{partial {A_{ij}}}}{{partial {x_j}}}} ,sum {frac{{partial {B_i}}}{{partial {x_i}}} in overline Omega } times [0,T] times R,left| {{B_i}} right| le Lambda ,left| C right| le Lambda ,],[left| {sum {frac{{partial {B_i}}}{{partial {x_i}}}} } right| le Lambda ,partial Omega in {C^2},v(r) in C[0,infty ).v(0) = 0,1 le frac{{rv(r)}}{{int_0^r {v(s)ds} }} le m,{u_0}(x) in {C^2}(overline Omega ),psi (s,t) in {C^beta }(partial Omega times [0,T]),0 < beta < 1],[{u_0}(s) = psi (s,0).]$ |
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