Method of upper and lower solutions and interval method for semilinear Euler-Poisson-Darboux equations (Q756017)

The method of descent and Duhamel's principle are used to deduce the solution formula for a linear equation, \(L_ cu=f(x,t)\), subject to zero Cauchy data, where \[ L_ cu\equiv u_{tt}+ku_ t/t-u_{xx}-c^ 2u. \] With this, the method of upper and lower solutions and the interval method are used to study existence and uniqueness of solutions, with or without the monotonicity requirement on the nonlinear term, for the semilinear Cauchy problem: \[ L_ ru=g(x,t;u)\text{ for } 0<t\leq T<\infty,\quad u(x,0)=0=u_ t(x,0). \] Here, k is a real parameter greater than or equal to 1, and r is a nonnegative constant.