Semilinear Hyperbolic Equations Research Articles

Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data

We study the initial boundary value problem of semilinear hyperbolic equations u t t − Δ u = f ( u ) u_{tt}-\Delta u=f(u) and semilinear parabolic equations u t − Δ u = f ( u ) u_{t}-\Delta u=f(u) with critical initial data E ( 0 ) = d E(0)=d (or J ( u 0 ) = d J(u_0)=d ), I ( u 0 ) > 0 I(u_0)>0 , and prove that there exist non-global solutions under classical conditions on f f .

Recent progress on controllability/observability for systems governed by partial differential equations

The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations. First, the authors review the theory for linear partial differential equations, including the iteration method for the null controllability of the time-invariant heat equation and the Rellich-type multiplier method for the exact controllability of the time-invariant wave equation, and especially a unified controllability/observability theory for parabolic and hyperbolic equations based on a global Carleman estimate. Then, the authors present sharp global controllability results for both semi-linear parabolic and hyperbolic equations, based on linearization approach, sharp observability estimates for the corresponding linearized systems and the fixed point argument. Finally, the authors survey the local null controllability result for a class of quasilinear parabolic equations based on the global Carleman estimate, and the local exact controllability result for general hyperbolic equations based on a new unbounded perturbation technique.

An augmented BV setting for feedback switching control

This paper considers dynamical systems under feedback with control actions limited to switching. The authors wish to understand the closed-loop systems as approximating multi-scale problems in which the implementation of switching merely acts on a fast scale. Such hybrid dynamical systems are extensively studied in the literature, but not much so far for feedback with partial state observation. This becomes in particular relevant when the dynamical systems are governed by partial differential equations. The authors introduce an augmented BV setting which permits recognition of certain fast scale effects and give a corresponding well-posedness result for observations with such minimal regularity. As an application for this setting, the authors show existence of solutions for systems of semilinear hyperbolic equations under such feedback with pointwise observations.

Trajectory and smooth attractors for Cahn–Hilliard equations with inertial term

This paper is devoted to a modification of the classical Cahn–Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient ε > 0, which is usually small in comparison with the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is ‘supercritical’ in the case of two and three space dimensions. Thus, the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small ε and arbitrary growth rate of the nonlinearity (see (Grasselli et al 2009 J. Evol. Eqns 9 371–404, Grasselli et al 2009 Commun. Partial Diff. Eqns 34 137–70)). The present contribution studies the long-time behaviour of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers (Grasselli et al 2009 J. Evol. Eqns 9 371–404, Grasselli et al 2009 Commun. Partial Diff. Eqns 34 137–70). In particular, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well. It is worth noting that the uniqueness of energy solutions in the 3D case is not known yet, so we have to use the so-called trajectory approach which does not require uniqueness. Finally, we apply the obtained exponential regularization of the energy solutions for verifying the dissipativity of solutions of the 2D modified Cahn–Hilliard equation in the intermediate phase space of weak solutions (in between energy and strong solutions) without any restriction on ε.

Regular Problems for Semilinear Hyperbolic Type Equations

In this paper we establish the wellposedness and regularity properties of solutions of Cauchy problems for semilinear hyperbolic equations of second order with unbounded principal operators. An example illustrating how our results apply is given.

Oscillatory properties for semilinear degenerate hyperbolic equations of second order

We consider three kind of oscillatory properties of the solutions to semilinear degenerate hyperbolic equations. Several sufficient conditions for the oscillation or non-oscillation are presented. In particular, they give us the positivity of the solutions for semilinear hyperbolic equations degenerating at initial point in one space dimension. Moreover we establish a few oscillatory conditions for the solutions of the mixed problem reduced to in one space dimension.

Variational optimality condition in the problem of control of initial boundary conditions for semilinear hyperbolic systems

Consideration is given to the problem of optimal control of a system of semilinear hyperbolic equations at finite-dimensional (pointwise) relations between initial boundary states of the system and control actions. In this case, the optimality condition in the form of a pointwise maximum principle is invalid. A nonclassical optimality condition of a variational type is proved. The sense of the obtained result is in that an optimal boundary or start control nearly at each point is a solution to a special problem of control of initial conditions for the system of ordinary differential equations constructed on the family of characteristics of the initial hyperbolic system. Constructions of the indicated optimality condition are illustrated by two examples. An iterative method based on the obtained result is stated. The method does not require additional assumptions about the differentiability of parameters of the control problem and convexity of the set of admissible controls necessary in using gradient methods.

Global existence for the one-dimensional second order semilinear hyperbolic equations

The aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one-dimensional semilinear equation appearing in the boundary value problems of gas dynamics. We investigate the Cauchy problem for such equation in the domain where the operator is weakly hyperbolic. We obtain the necessary condition for the existence of the self-similar solutions for the semilinear Gellerstedt-type equation. The approach used in the paper is based on the fundamental solution of the linear Gellerstedt operator and the L p – L q estimates.

Nonhomogeneous boundary value problem for a semilinear hyperbolic equation

We discuss the solvability of a nonhomogeneous boundary value problem for the semilinear equation of the vibrating string $x_{tt}(t,y)-\Delta x(t,y)+f(t,y,x(t,y))=0$ in a bounded domain and with a certain type of superlinear nonlinearity. To this end

A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincare Anal. Non Lineaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt. 46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl. 71 (1992) 455–467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.

Well‐posedness and long time behavior of a parabolic‐hyperbolic phase‐field system with singular potentials

AbstractIn this article, we study the long time behavior of a parabolic‐hyperbolic system arising from the theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation through an inertial term of the parabolic Allen–Cahn equation and it is characterized by the presence of a singular potential, e.g., of logarithmic type, instead of the classical double‐well potential. We first prove the existence and uniqueness of strong solutions when the inertial coefficient ε is small enough. Then, we construct a robust family of exponential attractors (as ε goes to 0). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Convergence of FEM with interpolated coefficients for semilinear hyperbolic equation

To solve spatially semidiscrete approximative solution of a class of semilinear hyperbolic equations, the finite element method (FEM) with interpolated coefficients is discussed. By use of semidiscrete finite element for linear problem as comparative function, the error estimate in L ∞ -norm is derived by the nonlinear argument in Chen [Structure theory of superconvergence of finite elements, Hunan Press of Science and Technology, Changsha, 2001 (in Chinese)]. This indicates that convergence of FEMs with interpolated coefficients for a semilinear equation is similar to that of classical FEMs.

Exact Controllability for Multidimensional Semilinear Hyperbolic Equations

In this paper, we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients. For this purpose, we establish an observability estimate for the linear hyperbolic equation with an unbounded potential, in which the crucial observability constant is estimated explicitly by a function of the norm of the potential. Such an estimate is obtained by a combination of a pointwise estimate and a global Carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem.

Blow-up of solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and boundary condition

Semilinear hyperbolic and parabolic initial–boundary value problems are studied. Criteria for solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and general boundary condition to blow up in finite time are obtained.

Global attractors for semilinear hyperbolic equations

In this paper, we use the generalized semiflow theory to study the longtime dynamical properties for a class of semilinear hyperbolic equations. The existence of global attractors is shown for the equations with no Lipschitz continuity assumption on their nonlinear terms. The results obtained here are generalizations of the related ones in Ball [Ball JM. Global attractors for damped semilinear wave equations. Discret Contin Dyn Syst 2004;10:31–52].

Global averaging and parametric resonances in damped semilinear wave equations

The long-time behaviour of solutions to a semilinear damped wave equation in a three-dimensional bounded domain with the nonlinearity rapidly oscillating in time (f = f(ε, u, t/ε)) is studied. It is proved that (under natural assumptions) the behaviour of solutions whose initial energy is not very large can be described in terms of global (uniform) attractors Aε of the corresponding dynamical processes and that, as ε → 0, these attractors tend to the global attractor A0 of the corresponding averaged system. We also give the detailed description of these attractors in the case where the limit attractor A0 is regular.Moreover, we give explicit examples of semilinear hyperbolic equations where the uniform attractor Âε (for the initial data belonging to the whole energy phase space) contains the irregular resonant part, which tends to infinity as ε → 0, and formulate the additional restrictions on the nonlinearity f which guarantee that this part is absent.

A PRIORIBOUNDS ON SPATIAL MOTIONS OF INCOMPRESSIBLE NONLINEARLY ELASTIC RODS

The geometrically exact quasilinear evolution equations governing the spatial motion of incompressible rods and their specializations to the hyperbolic equations governing nonlinearly elastic rods have novel mathematical structures strikingly different from those for the equations governing the motion of compressible rods. The main objectives of this paper are to formulate the governing equations, an exercise requiring the solution of a sequence of semilinear hyperbolic equations of first order, and to derive a priori estimates for certain strain variables, ensuring that they cannot reach geometrically prohibited ranges in finite time. This process exhibits a new system of physically important quasilinear equations worthy of careful analysis.

Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions

This article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term. Both dissipation and the source are fully nonlinear and the growth rate of the source map may include critical exponents (for Sobolev’s embedding H 1 → L 2 ). Besides continuity and monotonicity, no growth or regularity assumptions are imposed on the damping. We analyze the system in the presence of Neumann-type boundary conditions including the mixed cases: Dirichlet–Neumann–Robin. The damping affects a thin layer (a collar) near a portion of the boundary. To cope with the lack of control on the remaining section we develop a special method that accounts for propagation of the energy estimates from the dissipative region onto the entire domain. The Neumann system does not satisfy the Lopatinski condition in higher dimensions, hence the study of energy propagation in the absence of damping near the Neumann segment requires special geometric considerations.

Global behavior of solutions to an inverse problem for semilinear hyperbolic equations

This paper is concerned with global in time behavior of solutions for a semilinear, hyperbolic, inverse source problem. We prove two types of results. The first one is a global nonexistence result for smooth solutions when the data is chosen appropriately. The second type of results is the asymptotic stability of solutions when the integral constraint vanishes as t goes to infinity. Bibliography: 22 titles.

Homogenization for a nonlinear wave equation in domains with holes of small capacity

This paper is concerned with the homogenization of the nonlinear waveequation $\partial_{t t} u_{\varepsilon} - \Delta u_{\varepsilon} + \partial_t F(u_{\varepsilon}) = 0$ in $\Omega_{\varepsilon}\times(0,+\infty),$where $\Omega_{\varepsilon}$ is a domain containing holes withsmall capacity. In the context of optimal control,this semilinear hyperbolic equation was studied by Lions (1980)through a theory of ultra-weak solutions.Combining his arguments with the abstract frameworkproposed by Cioranescu and Murat (1982), for thehomogenization of elliptic problems, a new approach ispresented to solve the above nonlinear homogenization problem.In the linear case, one improves early classical resultsby Cionarescu, Donato, Murat and Zuazua (1991).