Solution regularity and smooth dependence for abstract equations and applications to hyperbolic PDEs

Abstract

In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but, mainly, we do not suppose that $F(\cdot,u)$ is smooth for all $u$. Even so, we state conditions such that for all $\lambda \approx 0$ there exists exactly one solution $u \approx u_0$, that $u$ is smooth in a certain abstract sense, and that the data-to-solution map $\lambda \mapsto u$ is smooth. In the second part we apply the results of the first part to time-periodic solutions of first-order hyperbolic systems of the type $$ \partial_tu_j + a_j(x,\lambda)\partial_xu_j + b_j(t,x,\lambda,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with reflection boundary conditions and of second-order hyperbolic equations of the type $$ \partial_t^2u-a(x,\lambda)^2\partial^2_xu+b(t,x,\lambda,u,\partial_tu,\partial_xu)=0, \; x\in(0,1) $$ with mixed boundary conditions (one Dirichlet and one Neumann). There are at least two distinguishing features of these results in comparison with the corresponding ones for parabolic PDEs: First, one has to prevent small divisors from coming up, and we present explicit sufficient conditions for that in terms of $u_0$ and of the data of the PDEs and of the boundary conditions. And second, in general smooth dependence of the coefficient functions $b_j$ and $b$ on $t$ is needed in order to get smooth dependence of the solution on $\lambda$, this is completely different to what is known for parabolic PDEs.