Existence Of Local Solutions For A Class Of Doubly Degenerate Parabolic Equations

In this paper,we consider the existence of local solution for a class of doubly degenerate parabolic equations with mixed boundary conditions.Diffusion equations.as an important class of parabolic equations,come from a variety of diffusion phenomena appeared widely in nature. They are suggested as mathematical models of physical problems in many fields such as filtration,phase transition and biochemistry.In many cases,the equations are nonlinear.Thus the study on the nonlinear problem is very important,it catches many mathe-maticians'eyes.Accordingly,the study on nonlinear problem has many contents and results,and some classical works about it can be found in the papers[1]-[7].Zhao[8]investigate the existence and nonexistence of solutions for the initial and boundary value problemwhere p > 2, Q_T = Ω× (0, T) and Ω∈ R~N is a bounded domain with smooth boundary (?)Ω.And Wang[9]consider the mixed boundary value problem for thep-Laplacian equation with sourcesut = div (|Vu|p~2Vu) + f(x, u, t), (x, t) G Qt,u(x, t) = 0, (x, t) G Ti x^ = 0, {x,t)€T2xwhere p > 2, Qt = ft x (0,T), 0 < T < +oo, Q is a bounded domain in RN with smooth boundary dQ, Tltr2 ^ 0 and Tj(jr2 = SflThey give the existence of local solution and nonexistence of global solution. Liu[10] discuss the existence of weak solutions of degenerate quasilinear parabolic equations initial boundary value problemNflifa, t, u, Du) + ao(x, t, u, Du) = f(x, t), (x, t) e QT,u(x,0) =on the space LP(0,T;V), where QT = Q, x (0,T), V = Wo>p(/x,Q) which is a weighted Sobolev space, /z(x) = (/ii(x),/X2(x),.. .,/x^(x)) and /Zt(z) > 0,i = 1,2,..., iV. Yin[ll]estabUsh the existence of continuous solutions of the first boundary value problem for the nonlinear diffusion equations of the formut = div(a(ti)|Vujm-xV?) + b(u)Vu, Qr = (0, T) x fl,?(a;lt) = Ol (x,t) e [0,T] xunder the conditions that A(s) = JJ)'a?^d N — 1, where N is the space dimension .Moreover,the uniqueness is proved for solutions with some regularities.In this paper.we consider the existence of local solution for a class of doubly degenerate parabolic equations with mixed boundary conditionsu(x,t) = 0,u(x,0) = uo(x), xeQ, (4)(*,t)GQt,(i)(?,*)r,x(o,n(2)(M)GF2x(o,n(3)where p > 2, QT = ft x (0,T), 0 < T < +oo, Q, is a bounded domain in Ti, F2 ^ 0, Fi (JF2 = 9fi, n denotes the outer unit normal on the boundary, uo(x)is smooth enough with uo{x) > 0, f(x, u, t) and a(u) satisfy(Hi) f{x, u, t) e Cx(fi x (0, +oo) x [0, T\), there exists a functiong(u) G ^((0, +oo)), such that f(x, u, t) < g(u), / > 0. (Ha) a(u) e Cx((0, +oo)), a{u) > 0,a'(u) > 0, o(0) = 0.Eq.(l)is of double degenerate type at the points where a(u) — 0 or Vu = 0. We assume thatA{u) = I a"-1 (s) ds,so we have/(x)W)<), (x,t)EQT, (5)?(*,?) = 0, (ar.tJerixCO.T), (6)^ = 0, (x,<)€rax(0,T)> (7)?(x,0) = ?o(a:), xen, (8)Owing to the degeneracy of Eq.(l),we consider the corresponding regular-ized problemut = div ((|VJ4(u)|2 + e)E^V/l(u)) +f(x,u,t), (x,t) € QTi (9)u(x,t)=e, (MJe^xfO.T), (10)^ = o, (M)er2x(o,T), (ii)u(x, 0) = uoe(x) + e, ien, (12)where 0 < e < 1, ?oe 6 C°°(^)> such that \\uoe +e||Loo(n) < \\u0 + l||i?>(fi), ||Vnoe||￡^(n) < ||Vuo||LP(n)) and uqc —* uq in Wl'v(n). It is well known that (9)-(12) has a classical solution ue, we first show u￡ > er.and by a'(u) > 0, we have a{ue) > a(e). Next we show some estimates on ue:ll?.||^?hi) < C, (13)\VA(uE)\pdxdt(QTl), and^^ G L2(QTl), and satisfyingo$?(x, 0) dx + // wyjtdasd^— //- ff \VA(u) \p-2VA{u) ? V(p dxdt = 0. JJqTiqTi for any (p € C°°(<5ri), which vanishes for t = Tx and (x,t) e Tx x (0,Ti).Finally, we prove the following theoremTheorem If (Hi) and (H2) hold.and u0 e ￡°°(fi)n#o(fi)> uo > 0, then there exists 7\ e [0, T], such that (1)—(4 )have a solution u on Qrt, satisfyingVA(u) e ISiQri, ^- e L\QTl).