Semilinear Hyperbolic Equations Research Articles

On a quasilinear parabolic–hyperbolic system arising in MEMS modeling

AbstractA coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.

Classical solution of the Cauchy problem for a semilinear hyperbolic equation in the case of two independent variables

In the upper half-plane, we consider a semilinear hyperbolic partial differential equation of order higher than two. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. The solution is constructed in an implicit analytical form as a solution of some integral equation. The local solvability of this equation is proved by the Banach fixed point theorem and/or the Schauder fixed point theorem. The global solvability of this equation is proved by the Leray-Schauder fixed point theorem. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established.

Determining a nonlinear hyperbolic system with unknown sources and nonlinearity

AbstractThis paper is devoted to some inverse boundary problems associated with a time‐dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. It is shown in several generic scenarios that one can uniquely determine the nonlinearity and/or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and an approximation property with higher order linearization for the semilinear hyperbolic equation.

Initial boundary value problem for semilinear hyperbolic equations with weakly damping term on conical manifolds with critical and supcritical initial data

Initial boundary value problem for semilinear hyperbolic equations with weakly damping term on conical manifolds with critical and supcritical initial data

Improved growth estimate of infinite time blowup solution for a semilinear hyperbolic equation with logarithmic nonlinearity

In this short paper, we give an improved estimate of the infinite time blowup solution for the semilinear wave equation with logarithmic nonlinearity. Comparing the previous work, this improved growth estimate not only provides the information about the relationship between the power of the nonlinearity and the growth, but also gives a faster growth estimate. The proof of the main conclusions is carried out at the negative initial energy level, the sub-critical initial energy level, the critical initial energy level and the arbitrary positive initial energy level.

Periodic Solutions of Quasi-Monotone Semilinear Multidimensional Hyperbolic Systems

This paper deals with the Cauchy problem for a class of first-order semilinear hyperbolic equations of the form ∂tfi+∑j=1dλij∂xjfi=Qi(f). where fi=fi(x,t) (i=1,⋯,n) and x=(x1,⋯,xd)∈IRd (n≥2,d≥1). Under assumption of the existence of a conserved quantity ∑iαifi for some α1,⋯,αn>0, of (strong) quasimonotonicity and an additional assumption on the speed vectors Λi=(λi1,⋯,λid)∈IRd—namely, span{Λj−Λk:j=1,⋯,n}=IRd for any k—it is proved that the set of constant steady state {f¯∈IRn:Q(f¯)=0} is asymptotically stable with respect to periodic perturbations, i.e., any initial data given by an periodic L1–perturbations of a constant steady state f¯ leads to a solution converging to another constant steady state g¯ (uniquely determined by the initial condition) as t→+∞.

Оптимальное управление каскадной системой гиперболических и обыкновенных дифференциальных уравнений с запаздыванием

In the class of smooth control functions, an optimal control problem of firstorder semilinear hyperbolic equations is investigated. We consider the case when the functional parameter in the right side of the hyperbolic system is determined from the controlled system of ordinary differential equations with constant state delay. Control functions are restricted by pointwise (amplitude) constraints. Problems of this kind arise when modeling a number of processes of population dynamics, interaction of a fluid (liquid or gas) with solids, etc. Optimal control methods based on the use of the Pontryagin maximum principle, its consequences and modifications are not applicable for such problems. The proposed approach is based on a special control variation, which ensures the smoothness of variable controls and the fulfillment of restrictions. The necessary optimality condition is proved. A scheme of a method for improving permissible control based on this condition is proposed, the convergence of the method is justified. An illustrative example is given.

Finite-time blowup and existence of global solutions for a logarithmic semilinear hyperbolic equation

In this paper we consider the semilinear wave equation with the product of logarithmic and polynomial nonlinearities and establish the global existence and finite-time blowup of solutions by using the potential well method.

Global well-posedness for a class of semilinear hyperbolic equations with singular potentials on manifolds with conical singularities

Global well-posedness for a class of semilinear hyperbolic equations with singular potentials on manifolds with conical singularities

Mathematical Analysis of Dynamic Models of Suspension Bridges with Delayed Damping

Suspension bridges are a type of construction in which the deck is suspended under a series of suspension cables that are on vertical hangers. The first modern example of this project began to appear in the early 1800s. Modern suspension bridges are lightweight, aesthetically pleasing and can span longer distances than any other bridge form. Many papers have been devoted to the modelling of suspension bridges, for instance, Lazer and McKenna studied the problem of nonlinear oscillation in a suspension bridge. They introduced a (one-dimensional) mathematical model for the bridge that takes into account of the fact that the coupling provided by the stays connecting the main cable to the deck of the road bed is fundamentally nonlinear, that is, they gave rise to the system of semi linear hyperbolic equation, where the first equation describes the vibration of the road bed in the vertical plain and the second equation describes that of the main cable from which the road bed is suspended by the tie cables. Recently, interest in this field has been increasing at a high rate. In this paper, we investigate some mathematical models of suspension bridges with a strong delay in linear aerodynamic resistance force. We establish the exponential decay of the solution for the corresponding homogeneous system and prove the existence of an absorbing set as well as a bounded attractor.

Age-structured delayed SIPCV epidemic model of HPV and cervical cancer cells dynamics I. Numerical method

The numerical method for simulation dynamics of nonlinear epidemic model of age-structured sub-populations of susceptible, infectious, precancerous and cancer cells and unstructured population of human papilloma virus (HPV) is developed (SIPCV model). Cell population dynamics is described by the initial-boundary value problem for the delayed semi-linear hyperbolic equations with age- and time-dependent coefficients and HPV dynamics is described by the initial problem for nonlinear delayed ODE. The model considers two time-delay parameters: the time between viral entry into a target susceptible cell and the production of new virus particles, and duration of the first stage of delayed immune response to HPV population growing. Using the method of characteristics and method of steps we obtain the exact solution of the SIPCV epidemic model in the form of explicit recurrent formulae. The numerical method designed for this solution and used the trapezoidal rule for integrals in recurrent formulae has a second order of accuracy. Numerical experiments with vanished mesh spacing illustrate the second order of accuracy of numerical solution with respect to the benchmark solution and show the dynamical regimes of cell-HPV population with the different phase portraits.

Time-domain decomposition for optimal control problems governed by semilinear hyperbolic systems with mixed two-point boundary conditions

Abstract In this article, we study the time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations and provide an in-depth well-posedness analysis. This is a continuation of our work, Krug et al. (2021) in that we now consider mixed two-point boundary value problems. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.

Discrete Schemes for a Class of Semilinear Hyperbolic Eqution

This paper focuses on the numerical algorithm of a class of semilinear hyperbolic equation. The dissipation term and external force source term existing in the equation enhance the nonlinearity of the model and make the nonlinear effect complicated. However, this nonlinear effect has a great influence on the discrete scheme, which cannot be neglected. Discretizing the nonlinear terms to ensure the validity of the scheme is the core issue in this paper. An effective scheme based on linearization techniques and iteration theory is proposed. It is based on the finite difference method. The efficiency of the proposed schemes was verified via some numerical examples showing that they compare well with existing methods.

On the solvability of a semilinear hyperbolic equation with a parameter

A semilinear hyperbolic equation with a parameter is considered. Sufficient conditions for unambiguous solvability of the boundary value problem are proved. A range of parameter values is specified for which a boundary value problem has a unique solution.

On the determination of nonlinear terms appearing in semilinear hyperbolic equations

We consider the inverse problem of determining the shape of a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary ( M , g ) of dimension n = 2 , 3 . We prove results of unique recovery of the nonlinear term F ( t , x , u ) , appearing in the equation ∂ t 2 u − Δ g u + F ( t , x , u ) = 0 on ( 0 , T ) × M with T > 0 , from partial knowledge of the solutions u on the lateral boundary ( 0 , T ) × ∂ M . We obtain, what seems to be, the first result of determination of the expression F ( t , x , u ) on the boundary x ∈ ∂ M for such a general class of nonlinear terms. With additional assumptions on the manifold and some extended measurements at t = 0 and t = T , we prove also the recovery of F inside the manifold x ∈ M .

Observer Design for a Class of Semilinear Hyperbolic PDEs With Distributed Sensing and Parametric Uncertainties

We study a class of heterodirectional semilinear hyperbolic partial differential equations, where the distributed state vector is partially measured. An observer estimating the unmeasured part of the state vector is designed. The design is, for instance, applicable to multiphase 1-D fluid models, where the pressure is measured, but the distributed flow and phase concentrations are not. Furthermore, the observer is extended to systems with parametric uncertainties appearing in the dynamics of the unmeasured part of the state. While required to be linear in the uncertain parameter, the uncertain term may be nonlinear in the state, even in the unmeasured part of the state. Terms of this type appear often in applications and cover, for instance, viscous drag in fluid flow systems. A noteworthy property of the design is that convergence of the state estimate is achieved without requiring persistent excitation. Two example applications are presented, and the design is illustrated in a simulation.

Задача оптимального управления гиперболической системой с запаздыванием на границе

The paper deals with an optimal control problem by a system of semilinear hyperbolic equations with boundary differential conditions with delay. This problem is considered for smooth controls. Because this requirement it is impossible to prove optimality conditions of Pontryagin maximum principle type and classic optimality conditions of gradient type. Problems of this kind arise when modeling the dynamics of non-interacting age-structured populations. Independent variables in this case are the age of the individuals and the time during which the process is considered. The functions of the process state describe the age-related population density. The goal of the control problem may be to achieve the specified population densities at the end of the process.The problem of identifying the functional parameters of models can also be considered as the optimal control problem with a quadratic cost functional. For the problem we obtain a non-classic necessary optimality condition which is based on using a special control variation that provides smoothness of controls. An iterative method for improving admissible controls is developed. An illustrative example demonstrates the effectiveness of the proposed approach.

Propagation of singularities for generalized solutions to nonlinear wave equations

The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof is based on a fixed point theorem in a suitable ultrametric topology on the subset of Colombeau solutions possessing the required regularity. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equations, and shows that the same behavior is attained in the case of non-classical, generalized solutions.

Regularity for Semilinear Neutral Hyperbolic Equations with Cosine Families

The purpose of this paper is to obtain the regularity for solutions of semilinear neutral hyperbolic equations with the nonlinear convolution. The principal operator is the infinitesimal generator of a cosine and sine families. In order to show a variation of constant formula for solutions, we make of using the nature of cosine and sine families.

Stability with respect to coefficients of solution of difference schemes approximating initial boundary-value problems for semi-linear hyperbolic equations

The stability with respect to coefficients of solution of a difference scheme approximating the initial boundary-value problem for the one-dimensional semi-linear hyperbolic equation is studied. The estimates of the solutions of both differential and difference problems are obtained. In the domain of existence of the solution, the estimates for perturbation of the solution of a difference scheme with respect to perturbation of the coefficients of the equation are obtained. These estimates are consistent with the estimates for the differential problem. In all cases, the method of energy inequalities, the Bihari inequality and its mesh analogue are used.