On the blow-up results for a class of strongly perturbed semilinear heat equations

Arch Rational Mech Anal

2017

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in some critical case, which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution.Mathematical subject classification: 35K57, 35K40, 35B44.

Section: Remark 14mentioning

confidence: 99%

Construction of a Blow-Up Solution for the Complex Ginzburg–Landau Equation in a Critical Case

Nouaili

Arch Rational Mech Anal

2017

“…Note also that in [12], the corresponding construction was done with ζ 0 = 0. Accordingly estimate (24) in [12] was satisfied with ζ 0 = 0 and φ 1,0 = 0. Here lays a major difference between our approach and that of [12], in the sense that we construct our solution in relation with (ζ i (s 0 ) + ζ 0 ) i , which is a particular solution of (12), whereas in [12], the construction is done only for ζ 0 = 0.…”

Section: Construction Of a Multi-soliton Solution In Similarity Variamentioning

confidence: 99%

“…By the way, the proof of (43) is far from being easy. Here, since we work with any ζ 0 ∈ Ê (see (24)) and aim at prescribing the center of mass, we don't need to be that accurate, and from this point of view, our proof is more simple than the proof of [12].…”

Section: Let Us Apply This Proposition Withmentioning

confidence: 99%

“…More generally, constructing a solution to some PDE with a prescribed behavior (not necessarily multi-solitons solutions) is an important question. That question was solved for (gKdV) by Côte [9,10], and also for parabolic equations exhibiting blow-up, like the semilinear heat equation by Bressan [3,4] (with an exponential source), Merle [43], Bricmont and Kupiainen [5], Merle and Zaag in [46,45], Schweyer [57] (in the critical case), Mahmoudi, Nouaili and Zaag [38] (in the periodic case), the complex Ginzburg-Landau equation by Zaag [62], Masmoudi and Zaag in [41] and also Nouaili and Zaag [55], a complex heat equation with no gradient structure by Nouaili and Zaag [54], a gradient perturbed heat equation in the subcritical case by Ebde and Zaag in [15], then by Tayachi and Zaag in the critical case in [58,59] and also by Ghoul, Nguyen and Zaag in [19], a strongly perturbed heat equation in Nguyen and Zaag [53], a non scaling invariant heat equation in Duong, Nguyen, and Zaag [14], two non variational parabolic system Ghoul, Nguyen and Zaag [20,21,22] or a higher order parabolic equation in [23]. Other examples are available for Schrödinger maps (see Merle, Raphaël and Rodnianski [44]), the wave maps (see Ghoul, Ibrahim and Nguyen [17]), and also for the Keller-Segel model (see Raphaël and Schweyer [56], and also Ghoul and Masmoudi [18]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Prescribing the center of mass of a multi-soliton solution for a perturbed semilinear wave equation

Hamza

Journal of Differential Equations

2019

We construct a finite-time blow-up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension. Given any integer k ≥ 2 and ζ 0 ∈ Ê, we construct a blow-up solution with a characteristic point a, such that the asymptotic behavior of the solution near (a, T (a)) shows a decoupled sum of k solitons with alternate signs, whose centers (in the hyperbolic geometry) have ζ 0 as a center of mass, for all times. Although the result is similar to the unperturbed case in its statement, our method is new. Indeed, our perturbed equation is not invariant under the Lorentz transform, and this requires new ideas. In fact, the main difficulty in this paper is to prescribe the center of mass ζ 0 ∈ Ê. We would like to mention that our method is valid also in the unperturbed case, and simplifies the original proof by Côte and Zaag [12], as far as the center of mass prescription is concerned. MSC 2010 Classification: 35L05, 35L71, 35L67, 35B44, 35B40, 35B20.

“…This two-step procedure has been successfully applied for various nonlinear evolution equations to construct both Type I and Type II blowup solutions. It was the case of the semilinear heat equation treated in [4], [32], [36] (see also [35], [9] for the case of logarithmic perturbations, [2], [3] and [16] for the exponential source, [37] for the complex-valued case), the Ginzburg-Landau equation in [27], [38] (see also [48] for an earlier work). It was also the nonlinear Schrödinger equation both in the mass critical [28,29,30,31] and mass supercritical [34] cases; the energy critical [10], [22] and supercritical [6] wave equation; the mass critical gKdV equation [24,25,26]; the two dimensional Keller-Segel model [42]; the energy critical and supercritical geometric equations: the wave maps [39] and [18], the Schrödinger maps [33] and the harmonic heat flow [40,41] and [17]; the semilinear heat equation in the energy critical [43] and supercritical [5] cases.…”

Section: Introductionmentioning

confidence: 99%

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

Ghoul

Nguyen

Advances in Pure and Applied Mathematics

2019

AbstractIn this note, we consider the semilinear heat system\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q>1,orf(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0.We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].