Initial-Boundary Value Problems For Two Classes Of Evolution Equations With A Weighted Integral Boundary Condition

Using functional analysis method, we will prove the existence and uniqueness of the generalized solution for equations(such as:a class of hyperbolic equation and a class of parabolic equation) with a weighted integral boundary condition. The method is based on an energy inequality and on the density of the range of the operator generated by the considered problem.Firstly, We study initial-boundary value problems for a class of hyperbolic equation.((?)2u)/\((?)t2)-1/x(?)/((?)x)(x((?)u/(?)x)f(x,t),(x,t)∈(0,1)×(0,T), u(x,0)=χ(x),u1(x,0)=ψ(x),x∈(0,1) u(x)(1,t)＝0,∫0αρ(x)u(x,t)dx+∫β1ρ(x)u(x,t)dx=0, t∈(0,T), where <α≤β<1,α+β=1and f(x,t),φ(x),ψ(x) are known functions. We prove the existence and uniqueness of the generalized solution using functional analysis method. Secondly,we study mixed problem f.0r a class of parabolic equation ut-1/(ρ(x))(ρ(x)ux)x-1/(ρ(x))(ρ(x)ux)xt=f(x,t),(x,t)∈(0,1)×(0,T), u(x,0)=φ(x), ux(1,t)=0t∈(0,T),∫0αρ(x)u(x,t)dx+∫β1ρ(x)u(x,t)dx=0, t∈(0,T), where0≤ρ(x)≤1,ρ(0)=0,0<α≤β<1, α+β=1and f (x,t),φ(x)are known functions.We prove the existence and uniqueness of the generalized solution.