Construction and stability of blowup solutions for a non-variational semilinear parabolic system

Ghoul, Tej-Eddine; Nguyen, Van Tien; Zaag, Hatem

doi:10.48550/arxiv.1610.09883

Cited by 4 publications

(28 citation statements)

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“…Note that the method of [24] has been proved to be successful for various situations including parabolic and hyperbolic equations. For the parabolic equations, we would like to mention the works by Masmoudi and Zaag [25] (see also the earlier work by Zaag [38]) for the complex Ginzburg-Landau equation with no gradient structure, by Nguyen and Zaag [27], [28] for a logarithmically perturbed nonlinear heat equation and for a refined blowup profile for equation (1.11), or by Nouaili and Zaag [26] for a non-variational complex-valued semilinear heat equation, by Ghoul, Nguyen and Zaag [17] for a non-variational parabolic system. There are also the cases for the construction of multi-solitons for the semilinear wave equation in one space dimension by Côte and Zaag [8].…”

Section: Introductionmentioning

confidence: 99%

Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

Ghoul

Nguyen

Zaag

2017

Journal of Differential Equations

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We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term:We construct for this equation a solution which blows up in finite time T > 0 and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time T simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some senses, resulting in the change of the final blowup profile in comparison with the case α = 0. The proof of the construction inspired by the method of Merle and Zaag in 1997, relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the blowup region, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations of the initial data.2010 Mathematics Subject Classification. Primary: 35K58, 35K55; Secondary: 35B40, 35B44.

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Section: Introductionmentioning

confidence: 99%

Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

Ghoul

Nguyen

Zaag

2017

Journal of Differential Equations

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“…This result improves a result by Friedman and Giga [16] where the method requires a very restrictive conditions p = q and µ = 1 in order to apply the maximum principle to suitable linear combination of the components u and v. The authors of [22] also derive the lower pointwise estimates on the final blowup profiles: for all 0 < |x| ≤ ǫ 1 , |x| 2(p+1) pq−1 u(T, x) ≥ ǫ 0 and |x| 2(q+1) pq−1 v(T, x) ≥ ǫ 0 , (1.11) for some ǫ 0 , ǫ 1 > 0. Recently, we establish in [18] the existence of finite time blowup solutions verifying the asymptotic behavior (1.10). In particular, we exhibit stable finite time blowup solutions according to the dynamics:…”

Section: Previous Literature and Statement Of The Resultsmentioning

confidence: 96%

“…The method we used in [18] is an extension of the technique developed by Merle and Zaag [25] treated for the standard semilinear heat equation…”

Section: Previous Literature and Statement Of The Resultsmentioning

confidence: 99%

Blowup solutions for a reaction–diffusion system with exponential nonlinearities

Ghoul

Nguyen

Zaag

2018

Journal of Differential Equations

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We consider the following parabolic system whose nonlinearity has no gradient structure:in the whole space R N . We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem.

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“…• The control of the finite dimensional problem thanks to a topological argument based on index theory. Note that this kind of topological arguments has proved to be successful also for the construction of type I blowup solutions for the semilinear heat equation (1.16) in [4], [48], [52] (see also [51] for the case of logarithmic perturbations, [5], [6] and [27] for the exponential source, [50] for the complex-valued case), the Ginzburg-Landau equation in [49] (see also [63] for an earlier work), a non-variational parabolic system in [28] and the semilinear wave equation in [15]. Note also that here we don't use the topological argument because the blow-up is stable.…”

Section: Introductionmentioning

confidence: 99%

On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

2019

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We will study the problem under an additional assumption of 1-corotational symmetry, with the following corotational ansatz Φ(x, t) = cos(u(|x|, t))x |x| sin(u(|x|, t)) 2 √ κ , where κ is an upper bound on the sectional curvature of the target manifold M (see Jost [31] and Lin-Wang [33]). Without these assumptions, the solution u(r, t) may develop singularities in some finite time (see for examples, Coron and Ghidaglia [14], Chen and Ding [11] for d ≥ 3, Chang, Ding and Yei [12] for d = 2). In this case, we say that u(r, t) blows up in a finite time T < +∞ in the sense that lim t→T ∇u(t) L ∞ = +∞.Here we call T the blowup time of u(x, t). The blowup has been divided by Struwe [60] into two types: u blows up with type I if: lim sup t→T (T − t) 1 2 ∇u(t) L ∞ < +∞, u blows up with type II if: lim sup t→T (T − t) 1 2 ∇u(t) L ∞ = +∞. 1-COROTATIONAL HARMONIC MAP HEAT FLOW IN SUPERCRITICAL DIMENSIONS 4 d−4−2 √ d−1 for d ≥ 11, correspond to the cases d = 2 and d = 7 in the study of equation (1.4) respectively.

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Construction and stability of blowup solutions for a non-variational semilinear parabolic system

Cited by 4 publications

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Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

Blowup solutions for a reaction–diffusion system with exponential nonlinearities

On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

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