scholarly journals Finite approximate controllability for semilinear heat equations in noncylindrical domains

2004 ◽  
Vol 76 (3) ◽  
pp. 475-487
Author(s):  
Silvano B. de Menezes ◽  
Juan Limaco ◽  
Luis A. Medeiros
Keyword(s):  
Fixed Point ◽  
Heat Equation ◽  
Approximate Controllability ◽  
State Equation ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
Fixed Point Result ◽  
Linearized Problem

We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.

scholarly journals Approximate controllability of the semilinear heat equation

1995 ◽  
Vol 125 (1) ◽  
pp. 31-61 ◽  
Author(s):  
Caroline Fabre ◽  
Jean-Pierre Puel ◽  
Enrike Zuazua
Keyword(s):  
Fixed Point ◽  
Heat Equation ◽  
Bounded Domain ◽  
Variational Approach ◽  
Approximate Controllability ◽  
Nonempty Subset ◽  
Linear Equations ◽  
Fixed Point Method ◽  
Controllability Problem ◽  
Semilinear Heat Equation

This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

scholarly journals Controllability of the Semilinear Heat Equation with Impulses and Delay on the State

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Hugo Leiva
Keyword(s):  
Continuous Function ◽  
Characteristic Function ◽  
Heat Equation ◽  
Bounded Domain ◽  
Distributed Control ◽  
Approximate Controllability ◽  
Nonempty Subset ◽  
Nonlinear Functions ◽  
Semilinear Heat Equation ◽  
Approximately Controllable

AbstractIn this paper we prove the interior approximate controllability of the following Semilinear Heat Equation with Impulses and Delaywhere Ω is a bounded domain in RN(N ≥ 1), φ : [−r, 0] × Ω → ℝ is a continuous function, ! is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ! and the distributed control u be- longs to L2([0, τ]; L2(Ω; )). Here r ≥ 0 is the delay and the nonlinear functions f , Ik : [0, τ] × ℝ × ℝ → ℝ are smooth enough, such thatUnder this condition we prove the following statement: For all open nonempty subset ! of Ω the system is approximately controllable on [0, τ], for all τ > 0.

The Cauchy problem for a system of nonlinear heat equations in two space dimensions

2015 ◽  
Vol 08 (01) ◽  
pp. 1550004
Author(s):  
Amel Chouichi ◽  
Sarah Otsmane
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Dimensional Space ◽  
Local Existence ◽  
Energy Functional ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Uniqueness Of Solutions ◽  
Semilinear Heat Equation ◽  
Singular Initial Data

This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity in two-dimensional space which is the analogue of the scalar model problem studied in [S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Local well posedness of a 2D semilinear heat equation, Bull. Belg. Math. Soc.21 (2014) 1–17]. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space (H1× H1)(ℝ2). The uniqueness part is nontrivial although it follows Brezis–Cazenave's proof [H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math.68 (1996) 73–90] in the case of monomial nonlinearity in dimension d ≥ 3. Next, we show that in the defocusing case our solution is bounded, and therefore exists globally in time. Finally, for this system, we treat the question of blow-up in finite time under the negativity condition on the energy functional. The technique to be used is adapted from [Bull. Belg. Math. Soc. 21 (2014) 1–17].

Convergence of blow-up solutions of nonlinear heat equations in the supercritical case

1999 ◽  
Vol 129 (6) ◽  
pp. 1197-1227 ◽  
Author(s):  
J. Matos
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Convergence Theorem ◽  
Nonlinear Parabolic Equation ◽  
Blow Up ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
Nonlinear Parabolic ◽  
Nonlinear Heat Equations ◽  
Self Similar

In this paper, we study the blow-up behaviour of the radially symmetric non-negative solutions u of the semilinear heat equation with supercritical power nonlinearity up (that is, (N – 2)p> N + 2). We prove the existence of non-trivial self-similar blow-up patterns of u around the blow-up point x = 0. This result follows from a convergence theorem for a nonlinear parabolic equation associated to the initial one after rescaling by similarity variables.

Blow-up for quasilinear heat equations with critical Fujita's exponents

1994 ◽  
Vol 124 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Victor A. Galaktionov
Keyword(s):  
Heat Equation ◽  
Critical Exponent ◽  
Critical Case ◽  
Source Term ◽  
Blow Up ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
The Cauchy Problem ◽  
Fixed Constant ◽  
Image Position

We consider the Cauchy problem for the quasilinear heat equationwhere σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivitywith σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.

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