A Liouville theorem for vector-valued nonlinear heat equations and applications

By F. Merle, Hatem Zaag. 2000

115 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b6b2137a01999f44253237a3dd16a62041ba1c3f

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Citations in the corpus

Concentration Compactness for Critical Wave Maps (2009, unknown): “As in [31], [32] we set y = x 1 − t , s = − log(1 − t), 0 < t < 1 and W(y, s, 0) := U(x, t) = U(e−sy, 1 − e−s), 0 ≤ s < ∞ where U is a weak wave map as constructed in Proposition 10.”
Threshold and generic type I behaviors for a supercritical nonlinear heat equation (2011, background): “By the blow-up cirterion of [19] (see also [17, Lemma 2.”
On growth rate near the blowup surface for semilinear wave equations (2005, background): “, Giga and Kohn [5, 6, 7], and Merle and Zaag [15] for the semilinear heat equation, and see Merle and Raphael [12, 13, 14] for the critical nonlinear Schrödinger equation (NLS)).”
Blow-up profile for the complex Ginzburg–Landau equation (2008, methodology): “When β = 0 and δ 6= 0, there is no Lyapunov functional, and the method of [15] and [16] breaks down.”
Determination of the blow-up rate for a critical semilinear wave equation (2005, background): “[8] F. Merle and H. Zaag.”
Nonradial type II blow up for the energy-supercritical semilinear heat equation (2016, background): “In the series of work [8, 9, 10, 11, 12, 33, 34], the authors show that in the energy subcritical range 1 < p < d+2 d−2 all blow-up solution are of type I and match the constant in space solution (1.”
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation (2006, methodology): “We remark that a similar structure exists in the case of nonlinear heat equations, as has been used by Giga and Kohn [8] and others ([24]).”
Why are solitons stable (2008, unknown): “The idea of using Liouville theorems to control asymptotic behaviour also appeared slightly earlier in the context of nonlinear parabolic equations in the work of Merle and Zaag [63].”
Travelling Fronts and Entire Solutions¶of the Fisher-KPP Equation in ℝN (2001, background): “In particular, there are no “pulse-like” solutions of (1), i.e., solutions such that u(x, t0) → 0a s |x |→+ ∞at some time t0 (see similar results for entire solutions of another class of parabolic equations in [ 25 ]).”
Profile for a Simultaneously Blowing up Solution to a Complex Valued Semilinear Heat Equation (2013, background): “tial data has been proved by Fermanian Kammerer, Merle and Zaag in [FKZ00] and [FKMZ00]. For more results on equation (3), see [Bal77], [GK85], [GK87], [GK89], [HV93], [HV94], [MM04], [MM09], [MZ98], [MZ00], [Miz07] and [QS07]. 2 As we inferred above, equation (1) appears as a complex generalization”
On Nonexistence of type II blowup for a supercritical nonlinear heat equation (2004, background): “Merle and Zaag [24, 25] refined the above results by proving that (1.7) ‖u(t)‖L∞ g(t) → 1 as t → T .”
Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation (2008, background): “A Liouville Theorem proved in [15] and [14] was crucially needed for the proof of the regularity result in the heat equation.”
Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view (2000, methodology): “u can have different types of behavior close to a blow-up point a. For example, from [HV92], [Vel92], [BK94] and [ MZ00 ], we have solutions such that”