The BV Solutions For One Order Quasilinear Equation With Borel Measure As Initial Condition

In this paper we consider the problem of the formin (x,t) QT with initial condition(2)for all x R, where m > 1 , 0 < p 1 is given real number,and is a non-negative -finite Borel measure in R = .By BV(QT), we mean the class of all functions in QT of locally bounded variation. In other words, u BV(QT) if and only if u Lloc1(QT) and uf and ux are regular measures in QT of locally bounded variation. By BV(R), we mean the class of all functions in R of locally bounded variation.Clearly,equation (1)-(2) has no classical solution in general.we considerits local BV solution.Definition1, A non-negative function u : QT (0,+ )issaid to be a solutin of (1). if u satisfies the following conditions [H1] and [H2]:[H1] For all R (0, + ), s (0,T),we have[H2] for V C0(QT) and 0, we havewheresign(u - k) = Definition 2. A non-negative function u : QT (0, + )issaid to be a solutin of (1)-(2). if u is a solution of (1) and satifiesthe initial condition (2) the following sense:ess lim our main results is the following theorem. Theorem 1. Let 0 < p 1 and be a non-negative -fmite Borel measure satifying the following growth condition.then Cauchy problem (l)-(2) has at least a solution u QT( ) sucn that R > r. (4)(5) for a.e.t (0,T,, ( )), where1, 2, 3 are positive depending on m,p and m > 1.0 < p 1.F =First we consider poroblem of the formwhere u0(x) L (R) BV(R) is a non-negative functon,fk(s) = min{sp, k}, k R, m > 1,0 < p < 1.because of existence of following equation,we get the existence of probleme (6).whereJ satisfies suppJ C (-,+ ), J(x)dx = 1. J(x) C0X(R).JF At last .we need the following propositon.Propositon1. Assume that m > 1,0 < p 1.r (0,+ ),u is solution of Cauchy problem (6).then there exist positive constant 1, 2, 3 depending m,p such thatfor a.e.t (0,Tr(u0)), wherePropositon2. Asumme that r (0,+ ),l (0.+ ) and u is solution of Cauchy problem(6),then there exist positive constant 4 depending m,p such thatess sup a.e.t (0,+ ), where m > 1.0 < p 1.Propositions, let r (0,+ ). u be solution of Cauchy problem (6), then, we haveR (r+x).and T (0, Tr(u0)).where QT(R) = (-R,+R) (T/2,T).and 5 is positive constant depending m,p.