Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation

Duong, Giao Ky; Nguyen, Van Tien; Zaag, Hatem

doi:10.2140/tunis.2019.1.13

Cited by 27 publications

(31 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1. 4) In addition to that, in [13], Herrero and Velázquez derived the same result with a different method. Particularly, in [17], Merle and Zaag gave a proof which is simpler than the one in [1] and proposed the following two-step method (see also the note [15]):…”

Section: Ealier Workmentioning

confidence: 62%

See 1 more Smart Citation

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

Duong

2019

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Ealier Workmentioning

confidence: 62%

“…and also the work of Duong, Nguyen and Zaag in [4], who considered the following non scale invariant equation ∂ t u = ∆u + |u| p−1 u ln α (2 + u 2 ).…”

Section: Ealier Workmentioning

confidence: 99%

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

Duong

2019

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let us mention that the blow-up question for the semilinear heat equation ∂ t u = ∆u + |u| p−1 u log a (2 + u 2 ) is studied by Duong-Nguyen-Zaag in [18]. More precisely, they construct for this equation a solution which blows up in finite time T , only at one blow-up point a, according to the following asymptotic dynamics:…”

Section: Introductionmentioning

confidence: 99%

The blow-up rate for a non-scaling invariant semilinear wave equations

Hamza

Zaag

2020

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

We consider the semilinear wave equationwith f (u) = |u| p−1 u log a (2 + u 2 ), where p > 1 and a ∈ R. We show an upper bound for any blow-up solution of (1). Then, in the one space dimensional case, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely u ′′ = |u| p−1 u log a (2+u 2 ). Unlike the pure power case (g(u) = |u| p−1 u) the difficulties here are due to the fact that equation (1) is not scale invariant.

show abstract

“…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

Zaag

2019

Advances in Pure and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

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].

show abstract

Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation

Cited by 27 publications

References 19 publications

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

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

The blow-up rate for a non-scaling invariant semilinear wave equations

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

Contact Info

Product

Resources

About