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Irina Kmit

Publications and source records attributed to Irina Kmit.

At least 19 recordsLinked to original sources
arXiv2026

Higher Regularity of Time-Periodic Solutions to Nonautonomous Hyperbolic Problems: Away from Resonances

We study higher regularity and its relation to nonresonant behavior for time-periodic solutions of boundary value problems for one-dimensional linear and nonlinear nonautonomous first-order integro-differential strictly hyperbolic systems. The boundary conditions include integral operators and various types of boundary reflections. We prove that continuous and classical solutions have $C^k$-regularity, provided the coefficients are sufficiently smooth and a suitable number of nonresonance conditions is satisfied. In the linear case, these conditions involve the principal coefficients, the diagonal lower-order coefficients, and the boundary reflection coefficients. In the nonlinear case, they also depend on the nonlinearities and on the solution itself. For nonautonomous hyperbolic systems, higher regularity generally requires additional nonresonance conditions, whose number depends on the desired order of differentiability. These conditions are not only sufficient but, in general, also necessary, revealing a distinctive feature of nonautonomous hyperbolic PDEs. By contrast, in the autonomous case, a single nonresonance condition (if one is needed at all) suffices to obtain arbitrarily high regularity. We also identify a class of nonautonomous hyperbolic problems for which no nonresonance conditions are required. In this case, the higher regularity of solutions is determined solely by the regularity of the data. The main technical tool underlying the proofs is an abstract regularity principle formulated in the setting of vector spaces.

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arXiv2026

An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation

This paper investigates an inverse boundary value problem for a semilinear strongly damped wave equation with Dirichlet boundary conditions in Sobolev spaces of functions bounded in time on $\R$, including periodic and almost periodic functions. In addition to constructing a bounded strong solution, we determine a time-dependent source coefficient via an integral overdetermination condition ensuring well-posedness. After reducing the inverse problem to a direct one, we first establish existence and uniqueness of solutions to an associated problem on finite time intervals. We then extend these solutions to half-lines and construct a bounded strong solution on the whole real line as a limit of such extensions, and subsequently establish its uniqueness. In particular, periodic and almost periodic data yield periodic and almost periodic solutions.

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arXiv2024

Lyapunov function and smooth periodic solutions to quasilinear 1D hyperbolic systems

We apply a Lyapunov function to obtain conditions for the existence and uniqueness of small classical time-periodic solutions to first order quasilinear 1D hyperbolic systems with (nonlinear) nonlocal boundary conditions in a strip. The boundary conditions cover different types of reflections from the boundary as well as integral operators with delays. In the first step we use a Lyapunov approach to derive sufficient conditions for the robust exponential stability of the boundary value problems for a linear(ized) homogeneous problem. Under those conditions and a number of non-resonance conditions, in the second step we prove the existence and uniqueness of smooth time-periodic solutions to the corresponding linear nonhomogeneous problems. In the third step, we prove a perturbation theorem stating that the periodic solutions survive under small perturbations of all coefficients of the hyperbolic system. In the last step, we apply the linear results to construct small and smooth time-periodic solutions to the quasilinear problems.

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arXiv2023

On the bounded smooth solutions to exponentially stable linear nonautonomous hyperbolic systems

We investigate global bounded solutions of higher regularity to boundary value problems for a general linear nonautonomous first order 1D hyperbolic system in a strip. We establish the existence of such solutions under the assumption of exponential stability and certain dissipativity (or non-resonant) conditions. Our results demonstrate the existence and uniqueness of $C^1$- and $C^2$-bounded solutions, in particular, periodic and almost periodic solutions. The number of imposed dissipativity conditions is related to the regularity of solutions. This connection arises due to the nonautonomous nature of the hyperbolic system under consideration, in contrast to autonomous settings. In the general case of global bounded smooth solutions we assume exponential stability in $H^1$. In the case of periodic solutions, the weaker assumption of $L^2$-exponential stability proves to be sufficient.

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arXiv2023

Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable $x$ and $C^\infty$-smooth with respect to the unknown function $u$, then the solution is $C^\infty$-smooth with respect to the time variable $t$, and if the PDEs are $C^\infty$-smooth with respect to $x$ and $u$, then the solution is $C^\infty$-smooth with respect to $t$ and $x$. The same is true for appropriate weak solutions. Moreover, we show examples of time-periodic functions, which do not satisfy the non-resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not $C^\infty$-smooth, neither with respect to $t$ nor with respect to $x$, even if the PDEs are $C^\infty$-smooth with respect to $x$ and $u$. For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant equations.

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arXiv2022

1D Hyperbolic Systems with Nonlinear Boundary Conditions II: Criteria for Finite Time Stability

We investigate the finite time stability property of one-dimensional nonautonomous initial boundary value problems for linear decoupled hyperbolic systems with nonlinear boundary conditions. We establish sufficient and necessary conditions under which continuous or $L^2$-generalized solutions stabilize to zero in a finite time. Our criteria are expressed in terms of a propagation operator along characteristic curves.

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arXiv2022

Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs

This paper concerns the behavior of time-periodic solutions to 1D dissipative autonomous semilinear hyperbolic PDEs under the influence of small time-periodic forcing. We show that the phenomenon of forced frequency locking happens similarly to the analogous phenomena known for ODEs or parabolic PDEs. However, the proofs are essentially more difficult than for ODEs or parabolic PDEs. In particular, non-resonance conditions are needed, which do not have counterparts in the cases of ODEs or parabolic PDEs. We derive a scalar equation which answers the main question of forced frequency locking: Which time shifts of the solution to the unforced equation do survive under which forcing?

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arXiv2021

Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems

The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the "loss of smoothness" property of hyperbolic PDEs.

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arXiv2020

Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$ \partial^2_t u(t,x)- a(x,\lambda)^2\partial_x^2u(t,x)= b(x,\lambda,u(t,x),u(t-\tau,x),\partial_tu(t,x),\partial_xu(t,x)), \; x \in (0,1) $$ with smooth coefficient functions $a$ and $b$ such that $a(x,\lambda)>0$ and $b(x,\lambda,0,0,0,0) = 0$ for all $x$ and $\lambda$. We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to $t$ and $x$) and smooth dependence (on $\tau$ and $\lambda$) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution $u=0$, and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter $\tau$. To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics, and then we apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem to this system. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the "loss of derivatives" property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays $\tau$.

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arXiv2020

Finite Time Stabilization of Nonautonomous First Order Hyperbolic Systems

We address nonautonomous initial boundary value problems for decoupled linear first-order one-dimensional hyperbolic systems, investigating the phenomenon of finite time stabilization. We establish sufficient and necessary conditions ensuring that solutions stabilize to zero in a finite time for any initial $L^2$-data. In the nonautonomous case we give a combinatorial criterion stating that the robust stabilization occurs if and only if the matrix of reflection boundary coefficients corresponds to a directed acyclic graph. An equivalent robust algebraic criterion is that the adjacency matrix of this graph is nilpotent. In the autonomous case we also provide a spectral stabilization criterion, which is nonrobust with respect to perturbations of the coefficients of the hyperbolic system.

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arXiv2014

Time-Periodic Second-Order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence

The paper concerns the general linear one-dimensional second-order hyperbolic equation $$ \partial^2_tu - a^2(x,t)\partial^2_xu + a_1(x,t)\partial_tu + a_2(x,t)\partial_xu + a_3(x,t)u=f(x,t), \quad x\in(0,1) $$ with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coefficients $a$, $a_1$, and $a_2$) ruling out the small divisors effect, we prove the Fredholm alternative. Moreover, we show that the solutions have higher regularity if the data have higher regularity and if additional non-resonance conditions are fulfilled. Finally, we state a result about smooth dependence on the data, where perturbations of the coefficient $a$ lead to the known loss of smoothness while perturbations of the coefficients $a_1$, $a_2$, and $a_3$ do not.

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arXiv2014

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

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.

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arXiv2012

Smoothing effect and Fredholm property for first-order hyperbolic PDEs

We give an exposition of recent results on regularity and Fredholm properties for first-order one-dimensional hyperbolic PDEs. We show that large classes of boundary operators cause an effect that smoothness increases with time. This property is the key in finding regularizers (parametrices) for hyperbolic problems. We construct regularizers for periodic problems for dissipative first-order linear hyperbolic PDEs and show that these problems are modeled by Fredholm operators of index zero.

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arXiv2011

Fredholm Solvability of Periodic Neumann Problem for a Linear Telegraph Equation

We investigate the linear telegraph equation $$ u_{tt}-u_{xx}+2\mu u_t=f(x,t) $$ with periodic Neumann boundary conditions. We prove that the operator of the problem is modeled as a Fredholm operator of index zero in the scale of Sobolev-type spaces of periodic functions. This result extends to small perturbations of the equation where $\mu$ becomes variable and discontinuous or an additional zero-order term appears. We also show that the solutions to the problem are smoothing.

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arXiv2010

Fredholmness and Smooth Dependence for Linear Time-Periodic Hyperbolic System

This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type $$ \partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,...,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution-map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur. We show that perturbations of the coefficients $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients $a_j$ and $b_{jk}$, but they are new even in the case of strict hyperbolicity and of smooth coefficients.

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arXiv2009

Smoothing Solutions to Initial-Boundary Problems for First-Order Hyperbolic Systems

We consider initial-boundary problems for general linear first-order strictly hyperbolic systems with local or nonlocal nonlinear boundary conditions. While boundary data are supposed to be smooth, initial conditions can contain distributions of any order of singularity. It is known that such problems have a unique continuous solution if the initial data are continuous. In the case of strongly singular initial data we prove the existence of a (unique) delta wave solution. In both cases, we say that a solution is smoothing if it eventually becomes $k$-times continuously differentiable for each $k$. Our main result is a criterion allowing us to determine whether or not the solution is smoothing. In particular, we prove a rather general smoothingness result in the case of classical boundary conditions.

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