Existence And Uniqueness Of BV Solutions Of Quasi-Linear Hyperbolic Equation With Local Finite Borel Measures As Initial Conditions

In this paper we consider the quasi-linear hyperbolic equationin Q_t = M×(0, T) with initial conditionsfor all x∈R, where 0 < m < 1 , andμ(x) is a nonnegativeσfinite Borel measure inIt is well known that equation (1.1) has no classical solution in general, we consider its local BV solution.Definition 1 A non-negative function u : Q_t (?) (0, +∞) is said to be a solution of (1.1), if u satisfies the following conditions [H1] and [H2]:[H1] For all R∈(0, +∞), and all s and T with 0 < s < T < +∞, we have[H2] For any withφ≥0, we haveDefinition 2 A non-negative function u : Q_t (?) (0, +∞) is said to be a solutionof the Cauchy problem (1.1)-(1.2), if u is a solution of (1.1) and satisfies the initial condition (1.2) in the following sense: Our main results are the following theorems.Theorem 1 Letμbe a non-negativeσ-finite Borel measure, then the Cauchy problem (1.1)-(1.2)has a solution at least.Theorem 2 Letμbe a non-negativeσ-finite Borel measure with a positive low boundβ, and satisfying the following growth conditionwhere , for all x∈(0, +∞), L₁ is a positive constant not dependingon x, then the Cauchy problem (1.1)-(1.2) has a unique solution u in Q_t such thatfor a.e. t∈(0,T), r∈(0, +∞), whereγ1 andγ2 are positive constants depending only on m.In order to prove the existence of Theorem 1 and uniquess in Theorem 2, at first we consider the regularized problem, and get several estimates of the solutions, then we can get the existence and uniquess of (1.1)-(1.2).In this part, we consider the existence and uniqueness of the Cauchy problem of the following formwhere u₀(x)∈L^∞(R)∩BV(R) is a non-negative function and 0 < m < 1. At first we consider the regularized equations of the formwith initial conditions whereand J_εsatisfies supp, .From classical quasi-linear parabolic theory, (1.8)-(1.9) has a nonnegative smooth solutions u_δ,ε(x). By some ideas in [18], we can obtain some estimates on u_δ,ε(x), and from these estimates, we can select a subsequence of {u_δ,ε}, and for convenience, we still denote by {u_δ,ε}, such thatasε→0⁺, where . We can prove function u_δsatisfies Definition 1.1 and Definition 2.2,then u_δis a solution of (1.6)-(1.7). In order to obtain the theorem 1, we prove these following propositons.Proposition 1. Let us be a solution of the Cauchy problem(1.6)-(1.7), then we havein the sense of distribution, where k = 1/(1 - m).Propositon 2. Let u_δbe a solution of the Cauchy problem(1.6)-(1.7), then we havewhere C is a positive constant only depending on m,Ψ_R(t) and N_r are defined as followingwhereα= 1/(1 - m),ξ_r∈C₀^∞(R) are a number of non-negative functions such that Propositon 3. Let u_δbe a solution of the Cauchy problem(1.6)-(1.7), then we havewhereΨ_R(t) is defined in lemma 3.2,Φ_r(t) is defined as following:Propositon 4. Let r∈(0, +∞) and u_δbe a solution of the Cauchy problem(1.6)-(1.7). Then there exist positive constantγ1 andγ2 depending only on m such thatfor all R > r and a.e. t∈(0,T).Propositon 5. Let r∈(0, +∞) and u_δbe a solution of the Cauchy problem(1.6)-(1.7), then we havefor all R∈(r,+∞) and T∈(0,+∞), whereΩ_T(R) = (-R,R)×(T/2,T),γ3 is a positive constant depending only on m.In order to obtain the theorem 2, we need the following lemmas.Lemma 1 Let u and v be solutions of (1.1) in Q_t, then we havefor all withφ≥0, where a₊ = max{a; 0} for all a∈R.Lemma 2 The Cauchy problem (5.4)-(5.5) exist a unique solution u(x, t) such that u(x,t)≥βfor a.e. (x,t)∈Q_t. Lemma 3 let∑_T be the set of solutions of the Cauchy problem (5.4)-(5.5) in Q_t·uL,(x,t) be solutions of equation (1.1) with the following initial conditionLemma 4 Supposeμhas a positive low boundβ, let∑_T be the set of solutions of the Cauchy problem (1.1)-(1.2) in Q_T, for ,supp , v_L are solutions of (1.1) with the following initial conditionif u∈∑_T, thenLemma 5 Supposeμhas a positive low boundβand satisfies (1.3), let u and v be two solutions of the Cauchy problem (1.1)-(1.2) in Q_t, if u and v satisfy (1.4), (1.5) andthen u(x,t) = v(x,t) for a.e.(x,t)∈Q_t.