Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary

Bandle, Catherine; Marcus, Moshe

doi:10.1016/s0294-1449(16)30162-7

Cited by 109 publications

(105 citation statements)

References 5 publications

(4 reference statements)

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“…also [4]) that the derivatives of the large solutions in the direction of the inner normal satisfy ∂u ∂ρ = − 2 p − 1 γ …”

Section: Asymptotic Behaviour Of the Gradientsmentioning

confidence: 96%

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Asymptotic behaviour of large solutions of quasilinear elliptic problems

Bandle

2003

Zeitschrift f�r Angewandte Mathematik und Physik (ZAMP)

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Abstract. The paper deals with the large solutions of the problems u = u p and u = e u . They blow up at the boundary. It is well-known that the first term in their asymptotic behaviour near the boundary is independent of the geometry of the boundary. We determine the second term which depends on the mean curvature of the nearest point on the boundary. The computation is based on suitable upper and lower solutions and on estimates given in [4]. In the last section these estimates are used together with the P -function to establish the asymptotic behaviour of the gradients. Mathematics Subject Classification (2000). 35B45, 35J25, 35J60.Keywords. Nonlinear boundary value problems, upper and lower solutions, boundary blowup, maximal solutions.

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“…also [4]) that the derivatives of the large solutions in the direction of the inner normal satisfy ∂u ∂ρ = − 2 p − 1 γ …”

Section: Asymptotic Behaviour Of the Gradientsmentioning

confidence: 96%

“…We determine the second term which depends on the mean curvature of the nearest point on the boundary. The computation is based on suitable upper and lower solutions and on estimates given in [4]. In the last section these estimates are used together with the P -function to establish the asymptotic behaviour of the gradients.…”

mentioning

confidence: 99%

Asymptotic behaviour of large solutions of quasilinear elliptic problems

Bandle

2003

Zeitschrift f�r Angewandte Mathematik und Physik (ZAMP)

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“…(Compare [10,39].) To obtain polyhomogeneity of solutions, for compatible polyhomogeneous initial 4 More general results can be found in [22]. 5 A similar inequality for the scalar wave equation has been independently derived in [34].…”

Section: A Weighted Energy Inequality For a Class Of Symmetric Hyperbmentioning

confidence: 99%

Polyhomogeneous solutions of wave equations in the radiation regime

Chruściel¹,

Lengard²

2000

Journées Équations Aux Dérivées Partielles

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We study the "hyperboloidal Cauchy problem" for linear and semilinear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behaviour at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a λφ p nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behaviour, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary I + of the Minkowski space-time.

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“…See also Bandle and Marcus [4] and Diaz and Letelier [7], for example, for more general f . Moreover, if Ω has a smooth boundary, an estimate up to an arbitrarily finite order was established by Andersson, Chruściel and Friedrich [1] and Mazzeo [23].…”

Section: Introductionmentioning

confidence: 99%