Cauchy problem for evolutionary pseudodifferential equations with variable symbols (Q261684)

The authors consider the problem of finding the solution of the evolutionary equation \(\partial u(t,x)/\partial t+Au(t, x)=0\), \(x\in\mathbb R\), \(0\leq t\leq T\), satisfying the initial condition \(u(t, x)|_{t=0}=\varphi (x)\), \(\varphi\) is a continuous function bounded and even on \(\mathbb R\). An operator \(A\) is given for a sumbol \(a\) by the formula \((A\psi)(x)=F^{-1}_{B_{\sigma\to x}}[a(t, x;\sigma)F_{B_{x\to\sigma}}[\psi](\sigma)](x)\) where \(F_{B_{\nu}}\) is a the Bessel transform. The main result is the following theorem. The Cauchy problem is solvable in the class of bounded continuous and even functions on \(\mathbb R\) and a solution of this problem is given by relation \(u(t, x)=\int_0^{\infty}Z(t, x; 0,\xi)\varphi (\xi)\xi^{2\nu+1}\,d\xi\) for a fundamental solution \(Z\) of the problem.