Solutions faibles d’équations paraboliques quasilinéaires avec données initiales mesurés

Alaa, Nour Eddine

doi:10.5802/ambp.64

Cited by 50 publications

(16 citation statements)

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“…In both [1] and [32], the largeness condition on u 0 is (roughly speaking) given in terms of an L 2 -product of u 0 with the principal eigenfunction of the Laplacian. The condition that appears in our Theorem 1.2 is essentially the same as the one in [32].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

Quaas

Rodríguez

2018

Journal of Differential Equations

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We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data C 1 up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

Quaas

Rodríguez

2018

Journal of Differential Equations

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“…From the basis we used in our Galerkin approximation it is immediate that u k = u k = 0 on T * . Now, for τ > 0 we have, from lemma 4, that there exists a subsequence u k n which converges to u in L 2 weak ((τ, T ); W 4,2 weak ( )). Therefore by taking a sequence of τ n which converges to zero as n goes to infinity and using the usual diagonal procedure we can find a subsequence u k m such that for any τ > 0, u k m will converge weakly in L 2 ((τ, T ); W 4,2 ( )) to u.…”

Section: Lemma 4 Let τ and T Be Numbers Such That 0 < τ < T < T * mentioning

confidence: 94%

“…We need to choose r 1 such that s ∈ [1,2] and (p − 1)θ 1. We recall that p is subject to assumption (2.5).…”

Section: Lemma 3 Assume Thatmentioning

confidence: 99%

“…Notice that all of the terms appearing in equation (3.56) are in L 2 ((0, T ); E ). Since the function w is in L 2 ((0, T ); E), a well-known lemma from Lions-Magenes [23] (see also [32]) implies that the function w(t) L 2 is absolutely continuous and that d/dt w(t) 2 L 2 = 2 ∂w/∂t, w E . Therefore, by taking the action of equation (3.56) on w and integrating by parts we find that…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Finite-time singularity versus global regularity for hyper-viscous Hamilton–Jacobi-like equations

2003

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The global regularity for the two-and three-dimensional Kuramoto-Sivashinsky equations is one of the major open questions in nonlinear analysis. Inspired by this question, we introduce in this paper a family of hyper-viscous Hamilton-Jacobi-like equations parametrized by the exponent in the nonlinear term, p, where in the case of the usual Hamilton-Jacobi nonlinearity, p = 2. Under certain conditions on the exponent p we prove the short-time existence of weak and strong solutions to this family of equations. We also show the uniqueness of strong solutions. Moreover, we prove the blow-up in finite time of certain solutions to this family of equations when the exponent p > 2. Furthermore, we discuss the difference in the formation and structure of the singularity between the viscous and hyper-viscous versions of this type of equation.

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“…Remark When

q < 2 - \frac{N}{N + 1}

, it is known in the case of Laplace operator (see [, , ] and partial results in ) that solutions exist even in case of initial data which are bounded Radon measures. We also expect that the results of the present Section, and particularly the estimates of Theorem , could be adapted in order to deal with the more general case of measures as initial conditions, although this is beyond the purpose of this article.…”

Section: The Casementioning

confidence: 99%

Local and global time decay for parabolic equations with super linear first-order terms

Magliocca

Porretta

2018

Proc. London Math. Soc.

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We study a class of parabolic equations having first-order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well-posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi-integrability, contraction principles and truncation methods for weak solutions.

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Solutions faibles d’équations paraboliques quasilinéaires avec données initiales mesurés

Cited by 50 publications

References 1 publication

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

Finite-time singularity versus global regularity for hyper-viscous Hamilton–Jacobi-like equations

Local and global time decay for parabolic equations with super linear first-order terms

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