arXiv subjects

M. A. Hamza

Publications and source records attributed to M. A. Hamza.

3 recordsLinked to original sources

arXiv2021

Refined estimates of the blow-up profile for a strongly perturbed semilinear wave equations in one space dimension

We consider in this paper a class of strongly perturbed semilinear wave equations with a non-characteristic point in one space dimension, for general initial data. Working in the framework of similarity variables, in \cite {MZ} Merle and Zaag constructed an explicit stationary solution of the unperturbed problem and proved an exponential convergence to this family of solutions. If we follow the same strategy under our strongly perturbed equation we just obtain a polynomial convergence which is a rough estimate compared to the one obtained in the unperturbed problem. In order to refine this approximation, we constructed an implicit solution to the perturbed problem which approaches the stationary solutions of the unperturbed problem and we prove the exponential convergence to this prescribed blow-up profile.

math.AP↗

arXiv2019

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

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 $k\ge 2$ and $ζ_0 \in \mathbb{R}$, 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 $ζ_0 \in \mathbb{R}$. We would like to mention that our method is valid also in the unperturbed case, and simplifies the original proof by Côte and Zaag \cite{CZcpam13}, as far as the center of mass prescription is concerned.

math.AP↗

arXiv2015

The blow-up rate for strongly perturbed semilinear wave equations in the conformal case

We consider in this work some class of strongly perturbed for the semilinear wave equation with conformal power nonlinearity. We obtain an optimal estimate for a radial blow-up solution and we have also obtained two less stronger estimates. These results are achieved in three-steps argument by the construction of a Lyapunov functional in similarity variables and the Pohozaev identity.

math.AP↗