L-1-L-p estimates for radial solutions of the wave equation and application

Author(s)

Ebert, M. R. ; Kapp, R. A. ; Picon, T.

• Document type: Journal article
• Source: Annali di Matematica Pura ed Applicata; v. 195, n. 4, p. 1081-1091, AUG 2016.
• Web of Science Citations: 1

Abstract

It is well known that, for space dimension n > 3, one cannot generally expect L-1-L-p estimates for the solution of u(tt) - Delta u = 0, u(0, x) = 0, u(t) (0, x) = g(x), where (t, x) is an element of R+ x R-n. In this paper, we investigate the benefits in the range of 1 <= p <= q such that L-p-L-q estimates hold under the assumption of radial initial data. In the particular case of odd space dimension, we prove L-1-L-q estimates for 1 <= q < 2n/n-1 and apply these estimates to study the global existence of small data solutions to the semilinear wave equation with power nonlinearity vertical bar u vertical bar(sigma), sigma > sigma(c) (n), where the critical exponent sigma(c)(n) is the Strauss index. (AU)

• FAPESP's process: 13/20297-8 - Decay estimates for hyperbolic partial differential equations in the L^p-L^q framework
• Grantee: Marcelo Rempel Ebert
• Support type: Scholarships abroad - Research
• FAPESP's process: 13/17636-5 - A priori estimates for elliptic complexes and applications
• Grantee: Tiago Henrique Picon
• Support type: Research Grants - Young Investigators Grants