| In this paper we consider the following equationswhere{D = 0 < t < T,0 < x < L,0 < y <∞},with the following initial and boundary conditions:where m≥1,minU1(t,x) > 0,1-1/e << p< 1.To give a precise statement of the results,we use the following transformation:Then (1.1)-(1.2) is transformed into the following initial-boundary value problem: whereΩT ={((?),ξ,η)| <(?)< T,0<ξ< L,0 <η<1} andWe now define the weak solution to(1.4)-(1.6).Letwhere . clearly andDefinition 1. w∈BV(ΩT)∩L∞(ΩT) is said to be a weak solution to (1.4)-(1.6),if the following conditions are satisfied:1) wη∈L2(ΩT) and there exists a positive constants C > 0 such that,2) (?)φ∈C01(ΩT), w satisfies the following integral inequality:where V = w-1.Then we can define the weak solutions to (1.1)-(1.2):Definition 2. u(t,x,y),v(t,x,y)is said to be a weak solution to problem (1.1)-(1.2).if the following conditions are fulfilled:2.there exists a constant C > 0,such that3.lct . then w((?),ξ,η) is a solution of (1.4)-(1.6)in the sense of Definition 1. 4. v is a regular measure on D and satisfies:The following theorems are the main results of this paper :Theorem 1 Assume that are appropriately smooth andand there exists a constant C0 such thatThen there exists a weak solution for any to the problem (1.4)-(1.6) whereλ= 2e3 max U1 + max|A + B -A/(1-η)|.As an immediate corollary of Theorem 2.1,we obtain the local existence of the weak solutions to system (1.1)-(1.2)Theorem 2 There exists a local weak solution to the problem (1.1)-(1.2). In order to obtain the above results ,Consider the regularized problem of (1.4)-(1.6): whereU1,w0ε,w1ε,w2ε are appropriately smooth andIn order to prove the compactness of wε, we need the following estimates:Lemma 1. let (1.12) and (1.17) hold, The the solution of the problem (1.4)-(1.6) satisfiesLemma 2. V satisfieswhere C is independent ofε, -1 <β,0 <α< 1, V = wε-1.Lemma 3. The solutions wεof (1.4)-(1.6) satisfyC is independent ofεand t.
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