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 $${\partial }_t{u}_j+a_j(x,\lambda ){\partial }_x{u}_j+b_j(t,x,\lambda ,u)=0,x\in (0,1),j=1,\dots ,n$$ with reflection boundary conditions and of second-order hyperbolic equations of the type $${\partial }_t^{2}u-a(x,\lambda {)}^2{\partial }_x^{2}u+b(t,x,\lambda ,u,{\partial }_{t}u,{\partial }_{x}u)=0,x\in (0,1)$$ 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.

48 pages

References in corpus (3)

• Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
• Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
• Smoothing effect and Fredholm property for first-order hyperbolic PDEs

Cited by in corpus (5)

• Classical Bounded and Almost Periodic Solutions to Quasilinear First-Order Hyperbolic Systems in a Strip
• Fredholm Solvability of Time-Periodic Boundary Value Hyperbolic Problems
• Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay
• Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs
• Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs