Excluding blowup at zero points of the potential by means of Liouville-type theorems

Guo, Jong‐Shenq; Souplet, Philippe

doi:10.1016/j.jde.2018.06.025

Cited by 13 publications

(10 citation statements)

References 28 publications

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“…Namely, the Liouville-type theorem in [31] states that any ancient solution of (1.22) with self-similar temporal decay at −∞ must depend on the time-variable only. This is then used to show that blow-up solutions of (1.22) satisfy |u t − |u| p−1 u| ≤ ε|u| p + C ε (see also [33], [19], [22] for related results based on the Liouville theorem in [31]). The proof of Theorem 1.3 relies on Theorem 1.1, combined with suitable rescaling and compactness arguments.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton–Jacobi equations

Filippucci

Pucci

Souplet

2019

Communications in Partial Differential Equations

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We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations: ∆u + |∇u| p = 0 and ut = ∆u + |∇u| p , with p > 2 and homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which asserts that any solution has to be one-dimensional. This turns out to be an efficient tool to study the behavior of boundary gradient blow-up (GBU) solutions of the parabolic problem in general bounded domains of R n with smooth boundaries.Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior of the form uνν ∼ −u p ν , with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any GBU point, and moreover implies that the behavior in the tangential direction is more singular. A description of the space-time profile is also obtained. The ODE type behavior and its connection with the Liouvilletype theorem can be considered as an analogue of the well-known results of Merle and Zaag [31] for the subcritical semilinear heat equation, with the significant difference that for the latter, the ODE behavior is in the time direction (instead of the normal spatial direction).On the other hand, it is known that any GBU solution admits a weak continuation, under the form of a global viscosity solution. As another consequence, we show that these viscosity solutions generically lose boundary conditions after GBU. Namely, solutions without loss of boundary conditions after GBU are exceptional and can be characterized as thresholds between global classical solutions and GBU solutions which lose boundary conditions. This result, as well as the above GBU profile, were up to now essentially known only in one space-dimension.Finally, in the case of elliptic Dirichlet problems, we deduce from our Liouville theorem an optimal Bernstein-type estimate, which gives a partial improvement of a local estimate of P.-L. Lions [28].

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We are now in a position to give the proof of Theorem 1.3, by combining Propositions 3.1-5.2 and an appropriate rescaling argument. As mentioned in Remark 1.2, we shall adapt the strategy in [31] to our problem, also using some simplifications from [22].…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton–Jacobi equations

Filippucci

Pucci

Souplet

2019

Communications in Partial Differential Equations

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“…Returning to the original variables, we get the estimate (24) on u t . Estimate ( 23) is easily obtained by (24), equation (2), and (18). The proof is now complete.…”

Section: Proof Of Corollary 1 Let Us Writementioning

confidence: 74%

“…Consider first the case of q > q c . The local L q norm in |x| ≤ K −θ T (T −t) 1/2+θω may be readily estimated by (18) and the change of variable…”

Section: Proof Of Corollary 1 Let Us Writementioning

confidence: 99%

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Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

Mukai¹,

Seki²

2021

DCDS

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We are concerned with blow-up mechanisms in a semilinear heat equation:ut = ∆u + |x| 2a u p , x ∈ R N , t > 0, where p > 1 and a > −1 are constants. As for the Fujita equation, which corresponds to a = 0, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if N ≥ 11 andwhere T is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case a = 0. As a consequence, we obtain an example of solutions that blow up at x = 0, the zero point of potential |x| 2a with a > 0, and behave in non-self-similar manner for N > 10 + 8a. This last result is in contrast to backward self-similar solutions previously obtained for N < 10 + 8a, which blow up at x = 0.

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Radial equivalence and applications to the qualitative theory for a class of nonhomogeneous reaction‐diffusion equations

Iagar

Sánchez

2023

Math Methods in App Sciences

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Some transformations acting on radially symmetric solutions to the following class of nonhomogeneous reaction‐diffusion equations which has been proposed in a number of previous mathematical works as well as in several physical models, are introduced. We consider here , , , and , real exponents. We apply these transformations in connection to previous results on the one hand to deduce general qualitative properties of radially symmetric solutions and on the other hand to construct self‐similar solutions, which are expected to be patterns for the dynamics of the equations, strongly improving the existing theory. We also introduce mappings between solutions which work in the semilinear case .

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Excluding blowup at zero points of the potential by means of Liouville-type theorems

Cited by 13 publications

References 28 publications

A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton–Jacobi equations

A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton–Jacobi equations

Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

Radial equivalence and applications to the qualitative theory for a class of nonhomogeneous reaction‐diffusion equations

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