BV Solutions For A Class Of Quasilinear Hyperbolic Equations With Measures As Initial Conditions

In nature, many physical problems can be transformed into qusilinear hyperbolic equations (systems),such as gas dynamic problem and traffic problem.It is well known that the existence and uniqueness of solutions for quasilinear hyperbolic equations with bounded functions as initial values have studied by a number of authors ,including P.D. Lax.In the case a finite Borel measure as initial function,the existence and uniqueness of BV solutions for the following equations of the formhave been obtained first by T.P.Liu and M.Pierre [3],where φ : R →R is locally Lipshitz contionous with φ(0) = 0, R = (—∞,+∞).In addition,the existence, non-existence and uniqueness forwith the Dirac measure as initial value are discussed by F.R.Guarguaglini [4].Such results have been obtained by one of authors in [5,6,7] for the following equtionsandat + ~a7 + uP = 0) (10)In addition,the existence ,non-existence and uniqueness for the equationdu dum n ._ + __=0, (m>l)with a—finite Borel measures as initial conditions are considered by one of authors in [8].in particular, we find a initial growth condition^z f dfijMr = sup (r~^z f dfij < +00, (1.1)which is a necessary and sufficient condition for the above equation to have a BV solution in QT = R x (0,T) for some T E (0, +oo) (see [8]).After,Some results are obtained for the following equations of the form [9]in Qt with initial conditionu(x,0) = n(x).for x G R ,where m > \,p > 0 and fx is a nonnegative a—finite Borel measure.In this paper we shall consider following hyperbolic quasilinear equation of the form^ + -j￡T + tqvP = 0, (m>l,p>0,q>0). (1.2)in Qt with initial conditionu{x,0) = fi{x). (1.3)for x G R ,where m > l,p > 0, g > 0 and n is a nonnegative cr-finite Borel measure.By the uniform BFestimates of the solutions , we firstly prove that the Cauchy problem (1.2) — (1.3) has a unique solution for any initial value under m < p r \ A~R 'Then Cauchy Problem (1.2)-(1.3yilas'a solution u(x,t) such that(1.5)whereTT((i) = 6zM~^m~x'', T"(/i) = lim rr(u),r-4+00and $1, $2 and #3 are positive constants depending only on m.