Controllability of a parabolic system with a diffuse interface

Jérôme Le Rousseau; Matthieu Léautaud; Luc Robbiano

Displaying similar documents to “Controllability of a parabolic system with a diffuse interface”

Estimates of Green functions and their applications for parabolic operators with singular potentials

Lotfi Riahi (2003)

Colloquium Mathematicae

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We prove global pointwise estimates for the Green function of a parabolic operator with potential in the parabolic Kato class on a $C^{1, 1}$ cylindrical domain Ω. We apply these estimates to obtain a new and shorter proof of the Harnack inequality [16], and to study the boundary behavior of nonnegative solutions.

On admissibility for parabolic equations in ℝⁿ

Martino Prizzi (2003)

Fundamenta Mathematicae

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We consider the parabolic equation (P) $u_{t} - Δ u = F (x, u)$ , (t,x) ∈ ℝ₊ × ℝⁿ, and the corresponding semiflow π in the phase space H¹. We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H¹ are π-admissible in the sense of Rybakowski. If F(x,u) is asymptotically linear, under appropriate non-resonance conditions, we use Conley’s index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained...

Boundary estimates for certain degenerate and singular parabolic equations

Benny Avelin, Ugo Gianazza, Sandro Salsa (2016)

Journal of the European Mathematical Society

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We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$ -Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_{T}$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of...

Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales

Pernilla Johnsen, Tatiana Lobkova (2018)

Applications of Mathematics

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This paper is devoted to the study of the linear parabolic problem $ε \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x / ε, t / ε^{3}) \nabla u_{ε} (x, t)) = f (x, t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $ε$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic...

Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

Alessandra Lunardi (1998)

Studia Mathematica

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We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in $ℝ^{n}$ . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

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The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of $p$ -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on $ℝ^{N}$

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove the existence of global attractors for the following semilinear degenerate parabolic equation on $ℝ^{N}$ : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales

Tatiana Danielsson, Pernilla Johnsen (2021)

Mathematica Bohemica

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In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^{2} (0, T; H_{0}^{1} (Ω))$ , fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $ε^{p} \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x ε^{- 1}, x ε^{- 2}, t ε^{- q}, t ε^{- r}) \nabla u_{ε} (x, t)) = f (x, t)$ , where $0 < p < q < r$ . The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by $p$ , compared to the standard matching that gives rise...

Application of Rothe's method to a parabolic inverse problem with nonlocal boundary condition

Yong-Hyok Jo, Myong-Hwan Ri (2022)

Applications of Mathematics

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We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value $u_{0} \in H^{1} (Ω)$ is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when $u_{0} \in L^{2} (Ω)$ and the integral kernel in the nonlocal boundary condition is symmetric. ...

On the long-time behaviour of solutions of the p-Laplacian parabolic system

Paweł Goldstein (2008)

Colloquium Mathematicae

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Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be $¹_{l o c}$ , and in the variable exponent case, L² and $W^{1, p (x)}$ -weak.

Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

Karine Beauchard (2011-2012)

Séminaire Laurent Schwartz — EDP et applications

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The goal of this note is to present the results of the references [5] and [4]. We study the null controllability of the parabolic equations associated with the Grushin-type operator $\partial_{x}^{2} + {| x |}^{2 γ} \partial_{y}^{2}$ ( $γ > 0$ ) in the rectangle $(x, y) \in (- 1, 1) \times (0, 1)$ or with the Kolmogorov-type operator $v^{γ} \partial_{x} f + \partial_{v}^{2} f$ ( $γ \in {1, 2}$ ) in the rectangle $(x, v) \in 𝕋 \times (- 1, 1)$ , under an additive control supported in an open subset $ω$ of the space domain. We prove that the Grushin-type...

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

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Consider the nonlinear heat equation (E): $u_{t} - Δ u = {| u |}^{p - 1} u + b {| \nabla u |}^{q}$ . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C ₁ {(T - t)}^{- 1 / (p - 1)} {\leq | | u (t) | |}_{\infty} \leq C ₂ {(T - t)}^{- 1 / (p - 1)}$ . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_{t} - u_{x x} \geq u^{p}$ . More general inequalities of the form $u_{t} - u_{x x} \geq f (u)$ with, for instance, $f (u) = (1 + u) l o g^{p} (1 + u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

Yūki Naito (2006)

Banach Center Publications

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We consider a nonlinear parabolic system modelling chemotaxis $u_{t} = \nabla \cdot (\nabla u - u \nabla v)$ , $v_{t} = Δ v + u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

Lipschitz stability in the determination of the principal part of a parabolic equation

Ganghua Yuan, Masahiro Yamamoto (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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Let $y (h) (t, x)$ be one solution to $\partial_{t} y (t, x) - \sum_{i, j = 1}^{n} \partial_{j} (a_{i j} (x) \partial_{i} y (t, x)) = h (t, x), 0 < t < T, x \in Ω$ with a non-homogeneous term $h$ , and ${y |}_{(0, T) \times \partial Ω} = 0$ , where $Ω \subset ℝ^{n}$ is a bounded domain. We discuss an inverse problem of determining $n (n + 1) / 2$ unknown functions $a_{i j}$ by ${\partial_{ν} y (h_{ℓ}) |_{(0, T) \times Γ_{0}}$ , $y (h_{ℓ}) (θ, \cdot)}_{1 \leq ℓ \leq ℓ_{0}}$ after selecting input sources $h_{1}, . . ., h_{ℓ_{0}}$ suitably, where $Γ_{0}$ is an arbitrary subboundary, $\partial_{ν}$ denotes the normal derivative, $0 < θ < T$ and $ℓ_{0} \in ℕ$ . In the case of $ℓ_{0} = {(n + 1)}^{2} n / 2$ , we prove the Lipschitz stability in the inverse problem if we choose $(h_{1}, . . ., h_{ℓ_{0}})$ from a set $ℋ \subset {C_{0}^{\infty} ((0, T) \times ω)}^{ℓ_{0}}$ with an arbitrarily fixed subdomain $ω \subset Ω$ . Moreover we can take $ℓ_{0} = (n + 3) n / 2$ by making...

Existence results for some nonlinear parabolic equations with degenerate coercivity and singular lower-order terms

Rabah Mecheter, Fares Mokhtari (2023)

Mathematica Bohemica

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In this paper, we study the existence results for some parabolic equations with degenerate coercivity, singular lower order term depending on the gradient, and positive initial data in $L^{1}$ .

The initial value problem for parabolic equations with data in $L^{p} (R^{n})$

Eugene Fabes (1972)

Studia Mathematica

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On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations

P. Besala (1971)

Annales Polonici Mathematici

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Estimates of weak solutions to nondiagonal quasilinear parabolic systems

Dmitry Portnyagin (2005)

Annales Polonici Mathematici

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$L^{\infty}$ -estimates of weak solutions are established for a quasilinear non-diagonal parabolic system with a special structure whose leading terms are modelled by p-Laplacians. A generalization of the weak maximum principle to systems of equations is employed.