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...

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. ...

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...

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.

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

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We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_{t} \geq - {(- Δ)}^{β / 2} u - V (x) u + h (x, t) u^{p}$ , where ${(- Δ)}^{β / 2}$ , 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying ${V ₊ (x) \sim a | x |}^{- b}$ , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Guillermo Reyes, Juan Luis Vázquez (2006)

Journal of the European Mathematical Society

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In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{*}$ , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the...

An observability estimate for parabolic equations from a measurable set in time and its applications

Kim Dang Phung, Gengsheng Wang (2013)

Journal of the European Mathematical Society

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This paper presents a new observability estimate for parabolic equations in $Ω \times (0, T)$ , where $Ω$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $Ω$ and a subset of positive measure in $(0, T)$ . This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

Strong maximum and minimum principles for parabolic functional-differential problems with nonlocal inequalities $[u^{j} (t_{0}, x) - K^{j}] + Σ_{i} h_{i} (x) [u^{j} (T_{i}, x) - K^{j}] \leq (\geq) 0$

Ludwik Byszewski (1990)

Annales Polonici Mathematici

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