A note on semilinear heat equation in hyperbolic space

Bandle et al. [1] obtained a quite interesting result about a semilinear heat equation that the Fujita exponent relative to the whole hyperbolic space is just the same as that relative to bounded domain in Euclidean space, and, in addition, the properties of solutions are different in the critical exponent case. Our purpose is to answer an open problem proposed by Bandle et al. for the critical exponent case, and it, together with the one obtained by them, shows that the critical exponent case does belong to the non-blow-up case, which is completely different from the case in Euclidean space.     

Null boundary controllability for semilinear heat equations

We consider null boundary controllability for one-dimensional semilinear heat equations. We obtain null boundary controllability results for semilinear equations when the initial data is bounded continuous and sufficiently small. In this work we also prove a version of the nonlinear Cauchy-Kowalevski theorem.W. Littman was partially supported by NSF Grant DMS 90-02919. The results of this paper were presented by Yung-Jen Lin Guo at the P.D.E. seminar at the University of Minnesota on January 27, 1993 and by W. Littman at the First International Conference on Dynamics Systems and Applications held in Atlanta in May 1993.     

Note on the cost of the approximate controllability for the heat equation with potential

We prove the approximate controllability for the heat equation with potential with a cost of order e^c/ε when the target is in with a precision in L2(Ω) norm. Also a quantification estimate of the unique continuation for initial data in L2(Ω) of the heat equation with potential is established.     

Local boundary controllability for the semilinear plate equation

weijiu_liu@uow.edu.

The problem of local controllability for the semilinear plate equation with Dirichlet boundary conditions is studied. By making use of Schauder's fixed point theorem and the inverse function theorem, we prove that this system is locally controllable under a super-linear assumption on the nonlinearity, that is, the initial states in a small neighborhood of 0 in a certain function space can be driven to rest by Dirichlet boundary controls. Our super-linear assumption includes the critical exponent.     

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

We derive global Carleman estimates for one-dimensional linear parabolic equations t_∂±x_∂(cx_∂) with a coefficient of bounded variations. These estimates are obtained by approximating c by piecewise constant coefficients, cε, and passing to the limit in the Carleman estimates associated to the operators defined with cε. Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semilinear equations.     

Null controllability for a semilinear parabolic equation with gradient quadratic growth

This article is concerned with the null controllability of a semilinear parabolic equation with the nonlinear term involving the gradient quadratic term. The technique in this paper is a combination of Cole–Hopf transformation and some methods from [A.Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control Optim. 41 (2003) 1886–1900].     

A Liouville property and quasiconvergence for a semilinear heat equation

This paper is concerned with a supercritical semilinear diffusion equation with the power nonlinearity. Via establishing a Liouville-type property, we prove the quasiconvergence (convergence to a set of steady states) of a large class of global solutions. The method of proof relies on similarity variables and invariant manifold ideas.     

A Kalman-type condition for stochastic approximate controllability

We are interested in the approximate controllability property for a linear stochastic differential equation. For deterministic control necessary and sufficient criterion exists and is called Kalman condition. In the stochastic framework criteria are already known either when the control fully acts on the noise coefficient or when there is no control acting on the noise. We propose a generalization of Kalman condition for the general case. To cite this article: D. Goreac, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Refined asymptotic profiles for a semilinear heat equation

We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation,

Remarks on approximate controllability

Dedicated to Professor Shmuel Agmon     

Pullback attractors for a semilinear heat equation on time-varying domains

The existence of a pullback attractor is established for the nonautonomous dynamical system generated by the weak solutions of a semilinear heat equation on time-varying domains with homogeneous Dirichlet boundary conditions. It is assumed that the spatial domains Ot in RN are obtained from a bounded base domain O by a C²-diffeomorphism, which is continuously differentiable in the time variable, and are contained, in the past, in a common bounded domain.     

Approximate controllability for semilinear retarded systems

This paper deals with the approximate controllability for the semilinear retarded control system. We will also derive the equivalent relation between controllability and stabilizability of the solution for the corresponding linear control system.     

Exponential convergence for a periodically driven semilinear heat equation

We consider a semilinear heat equation in one space dimension, with a periodic source at the origin. We study the solution, which describes the equilibrium of this system and we prove that, as the space variable tends to infinity, the solution becomes, exponentially fast, asymptotic to a steady state. The key to the proof of this result is a Harnack type inequality, which we obtain using probabilistic ideas.     

Multiple blowup of solutions for a semilinear heat equation

The present paper is concerned with a Cauchy problem for a semilinear heat equation with u₀L(R^N). A solution u of (P) is said to blow up at t=T<+ if lim sup_tT|u(t)|=+ with the supremum norm |·| in R^N. We show that if and N11, then there exists a proper solution u of (P) which blows up at t=T₁, becomes a regular solution for t(T₁,T₂) and blows up again at t=T₂ for some T₁,T₂ with 0<T₁<T₂<+.Mathematics Subject Classification (2000): 35K20, 35K55, 58K57Revised version: 20 July 2004Acknowledgment The author expresses her gratitude to Professor Marek Fila for useful discussion.     

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C¹). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.