A blowup solution of a complex semi-linear heat equation with an irrational power

Abstract

In this paper, we consider the following semi-linear complex heat equation ∂tu = ∆u + u p , u ∈ C in R n , with an arbitrary power p, p > 1. In particular, p can be non integer and even irrational, unlike our previous work [5], dedicated to the integer case. We construct for this equation a complex solution u = u 1 + iu 2 , which blows up in finite time T and only at one blowup point a. Moreover, we also describe the asymptotics of the solution by the following final profiles:2) This model is connected to the viscous Constantin-Lax-Majda equation with a viscosity term, which is a one dimensional model for the vorticity equation in fluids. For more details, the readers are addressed to the following works: Constantin, Lax, Majda [2], Guo, Ninomiya and Yanagida in [8], Okamoto, Sakajo and Wunsch [25], Sakajo in [26] and [27], Schochet [28]. In [5], we treated the case p ∈ N. Indeed, handling the 2010 Mathematics Subject Classification. Primary: 35K50, 35B40; Secondary: 35K55, 35K57.

Refined asymptotics for the blow-up solution of the complex Ginzburg–Landau equation in the subcritical case

“…For parabolic equations, it has been used in [24] and [39] for the complex Ginzburg Landau (CGL) equation with no gradient structure, the critical harmonic heat flow in [31], the two-dimensional Keller-Segel equation in [32] and the nonlinear heat equation involving a nonlinear gradient term in [12,36]. Recently, this method has been applied to various nonvariational parabolic systems in [27] and [13][14][15][16], and to a logarithmically perturbed nonlinear equation in [7][8][9]26]. We also mention a result for a higher-order parabolic equation [17] and in [1,11] two more results for equations involving nonlocal terms.…”

Construction of blowup solutions for the Complex Ginzburg-Landau equation with critical parameters

“…For parabolic equations, it has been used in [MZ08] and [Zaa01] for the Complex Ginzburg Landau (CGL) equation with no gradient structure, the critical harmonic heat flow in [RS13], the two dimensional Keller-Segel equation in [RS14] and the nonlinear heat equation involving nonlinear gradient term in [EZ11], [TZ19]. Recently, this method has been applied for various non variational parabolic system in [NZ15] and [GNZ17, GNZ18b, GNZ18a, GNZ19], for a logarithmically perturbed nonlinear equation in [NZ16,Duo19b,Duo19a,DNZ19]. We also mention a result for a higher order parabolic equation [GNZ20], two more results for equation involving non local terms in [DZ19,AZ19].…”

Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation

“…Note that the constructive method given in those works was efficiently used in a very large class of parabolic equations such as in Merle and Zaag [53] for quenching problems; in Duong et al [18], Nguyen and Zaag in [57], and Tayachi and Zaag [67] for perturbed nonlinear source terms; in Duong et al [19,20], Masmoudi and Zaag [55] and Nouaili and Zaag [59] for the Complex Ginzburg-Landau equation; and Duong [26,27], and also in Nouaili and Zaag [56] for non-variational complex valued heat equations.…”

“…We also mention other situations where a clever adaptation of the shrinking set lead to the derivation of a sharper blowup behavior. This was in particular the case a complex-valued heat equation with no variational structure, where the behavior of the imaginary part was derived by Duong in [26,27]. We also mention the case of the Complex Ginzburg-Landau equation (CGL) in some critical setting in Duong et al [19] and Nouaili and Zaag [59]; and in the subcritical range in Duong et al [20] too, where a higher order expansion was derived thanks to a good adaptation of the shrinking set.…”