Existence Of BV Solutions For A Quasilinear Hyperbolic Equation With Dirac Measure As Source

In this paper we consider the problem for systems of quasilinear hyperbolic equation of the formwith the initial conditionwhere m > 1,u₀∈L¹(R).δ(x) is the Diriac measure centered at original point.Clearly, the Cauchy problem (1.1)-(1.2) has no classical solution in the general. Therefore we consider its BV solutions.Definition 1. A nonnegative function u : Q_T (?) (0, +∞) is said to be a solutin of (1.1), if u satisfies the following conditions [H₁] and [H₂]:[H1] For any R∈(0, +∞)and s∈(0, T), we havewe haveDefinition 2. A non-negative function u : Q_T (?) (0,+∞) is said to be a solutin of Cauchy problem (1.1)-(1.2), if u is a solution of (1.1) and satisfies the initial conditions (1.2) in the following sense : Our main result is the following theorem.Theorem 1.(Existence) Let m > 1, then the Cauchy problem (1.1) - (1.2) has at least one solution.At first we consider the regularized equations of the formwith initial and boundary conditionswhereand J_εsatisfy supp let compact set K (?)Q_T(R), from classical parabolic theorem, problem (3.3)-(3.5) has a smooth solution u_R, and u_0ε satisfy u_0ε≤A, Applying extremum principle , we get the maximum estimate,we havewhere C only depends on K.we can select a subsequence of {u_R}, and for convienience, we still denote by {u_R}, for (?)K(?)Q_T such that {u_r}→{u_h,ε in C^{2+α,1+α/2}, and u_h,ε be a solution of the regularized equation with initial conditionIn order to obtain the theorem 1, we prove the following lemmas and propositons.lemma 1 assume that m > 1 and let uis a solution of the Cauchy problem (4.1), then we havewherelemma 2 assume that u is a solution of the Cauchy problem (4.1), then we havefor a.e. t∈(0, +∞), wherelemma 3 assume that u is a solution of the Cauchy problem (4.1), then we havewhereΦ(·) andΨ(·) axe defined by (4.3), (4.4) andproposition 1 assume that m > l,r∈(0,+∞), u is a solution of the Cauchy problem (4.1), then there exist positive constantsγ1,γ2,γ3 depends only on m,such that for a.e. t∈(0,T_r(u₀)), whereproposition 2 assume that m > 1,r∈(0,∞),1∈(0,∞), and let u is a solution of the Cauchy problem (4.1), then there exist positive constant 74 depends only on m, such thatfor a.e.t∈(0, +∞).proposition 3 assume that r∈(0, +∞),and let uis a solution of the Cauchy problem (4.1), then we havefor all R∈(r,+∞) and T∈(0,T_r(u₀)). where Q_T(R) = (-R,R)×(T/2,T),γ5 is a positive depends only on m.