The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption

Article type: Research Article

Authors: Hamza, M.A.

Affiliations: Deanship of Preparatory and Supporting Studies at Dammam, University of Dammam, Dammam, Saudi Arabia. E-mail: mahamza@uod.edu.sa

Abstract: We consider in this paper a large class of perturbed semilinear wave equations with critical (in the conformal transform sense) power nonlinearity. We will show that the blow-up rate of any singular solution is given by the solution of the non-perturbed associated ODE. The result in the radial case has been proved in Math. Phys. Anal. Geom. 18(1) (2015), Art. 15. The same approach will be followed here, but the main difference is to construct a Lyapunov functional in similarity variables valid in the non-radial case, which is far from being trivial. That functional is obtained by combining some classical estimates and a new identity of the Pohozaev type obtained by multiplying Eq. (1.7) by y·∇w in a suitable weighted space.

Keywords: semilinear wave equation, blow-up, perturbations, conformal exponent, Pohozaev identity

DOI: 10.3233/ASY-161358

Journal: Asymptotic Analysis, vol. 97, no. 3-4, pp. 351-378, 2016