The BV Solution For A Class Of 2D Quasilinear Hyperbolic Equation With Measure As Initial Condition

In this paper we consider the following Cauchy problem for systems of quasilinear hyperbolic equation of the formwhere m,n >1,μ(x) is a non-negative finite Borel measure in R≡(-∞, +∞), v(y)∈L^∞(R)∩C¹(R).Clearly,the Cauchy problem (1.1)-(1.2) has no classical solution in the general,Therefore we consider its BV solutions.Our main results are the following theorems.Theorem 1.(Existence) Let 1 < n < m + 1, (?) dμ= M; v(y) > 0,v′(y)≥0,(?)y∈R,(?)ω_λ(y)v′(y)dy = M₁, (?)ω_λ(y)v(y)dy = M₂,ω_λ(y) = exp{-λ(?)},(?)λ> 0,M₁,M₂ are positive constants aboutλ,then the Cauchy problem (1.1)-(1.2) has a solution.Theorem 2.(Uniqueness) Let 1 < n < m + 1, (?) dμ=M; v(y) > 0,v′(y)≥0,(?)y∈R,(?)ω_λ(y)v'(y)dy = M₁, (?)ω_λ(y)v(y)dy = M₂,ω_λ(y) = exp{-λ(?)},(?)λ> 0,M₁,M₂ arepositive constants aboutλ,then the solution of the Cauchy problem (1.1)-(1.2) is unique.In this paper,we consider the following definition.Definition 1:A nonnegative function u :Q→[0, +∞) is said to be a solution of (1.1),if u satisfies the following conditions [A] and [B]: [A] :For any 0 <Ï„< T < +∞, (?)R∈(0, +∞),we have [B] :For anyφ∈C₀^∞(Q),andφ≥0,,we haveDefinition 2: A nonnegative function u :Q→[0, +∞) is said to be a solution of (1.1)-(1.2), if u is a solution of (1.1), and satisfies the initial conditions (1.2) in the sense of the distribution:At first we consider the regularized problemwhereand J_εsatisfyAndμ_ε(x), v(y) satisfy the following conditions:μ_ε(x)≥0, v(y) >0,v′(y)≥0,Where,M₂ is a positive constant aboutλ. From [2], above problem has a solutionμ_ε∈L^∞(Q). At First,we haveIn order to obtain the theorem 1,we prove the following propositions.Proposition 1. Let u_εbe a solution of the Cauchy problem (1.1)-(1.2) in Q.Then we havewhere,γ> 0 is a constant only about m.Proposition 2. Let u_ε∈L^∞(Q) be a solution of the Cauchy problem (1.1)-(1.2) in Q.Then we have∫∫_R²u_ε(x,y,t)ωλ(y)dxdy+∫∫∫_Q (?)u_εⁿ/(?)yωλ(y)dxdydÏ„=∫∫_R²Î¼_ε(x)v(y)ωλ(y)dxdy≤MM₂a.e. t∈(0,+∞),∫_Rμ_ε(x)dx≤∫_Rdμ=M,∫_Rv(y)ωλ(y)dy=M₂,,where,M₂ is apositive constant aboutλ.Proposition 3. Let u_εbe a solution of the Cauchy problem (1.1)-(1.2) in Q.Then we haveWhere, M₁,M₂ are positive constants aboutλ.Proposition 4. Let u_εbe a solution of the Cauchy problem (1.1)-(1.2) in Q.Then wehaveProposition 5. Let u_εbe a solution of the Cauchy problem (1.1)-(1.2) in Q.Then we haveWhere,M₂ is a positive constant aboutλ. Proposition 6. Let u_εbe a solution of the Cauchy problem (1.1)-(1.2) in Q.Then we haveWhere, M₂ is a positive constant aboutλ.In order to obtain the theorem 2,we prove the following lemmas.Lemma 1. Let u₁,u₂ be two solutions of the Cauchy problem (1.1)-(1.2) in Q.ifThen for a.e. (x, y, t)∈Q,we have u₁(x, y, t) = u₂(x, y, t).Lemma 2. Let∑be the set of solutions of (1.1)-(1.2).Letχ(x)∈C₀^∞(R),satisfying 0≤χ(x)≤1, w be a solution of (1.1).It satisfies the following initial condition.w(x,y,0) =χ(x)μ(x)v(y) For any (x, y)∈R²,if u∈∑, thenw(x,y,t)≤u(x,y,t),for a.e. (x,y,t)∈Q.Lemma 3. Let u∈∑,then for a.e.(x, y, t)∈Q,we haveLemma 4. Let u be a solution of (1.1)-(1.2), then we havefor a.e. t∈(0, +∞).Where, M₂ is a positive constant aboutλ.Lemma 5. Let u be a solution of (1.1)-(1.2), then we havefor a.e. t∈(0, +∞), (?)R>0,where,ε(R) =(?)ω_λ(y)v(y)dμdy.