A shift in the Strauss exponent for semilinear wave equations with a not effective damping

Author(s)

D'Abbicco, Marcello ^[1] ; Lucente, Sandra ^[2] ; Reissig, Michael ^[3]

Affiliation

^[1] Univ Sao Paulo, Dept Comp & Matemat, BR-14040901 Ribeirao Preto, SP - Brazil

^[2] Univ Bari, Dept Math, I-70125 Bari - Italy

^[3] Tech Univ Bergakad Freiberg, Fac Math & Comp Sci, D-09596 Freiberg - Germany

• Document type: Journal article
• Source: Journal of Differential Equations; v. 259, n. 10, p. 5040-5073, NOV 15 2015.
• Web of Science Citations: 26

Abstract

In this note we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with a not effective scale-invariant damping term, namely v(tt) - Delta v +2/1+t v(t) = broken vertical bar v broken vertical bar(P), v(0, x) = v(0)(x), v(t) (0, = v(1)(x), where p > 1, n > 2. We prove blow-up in finite time in the subcritical range p is an element of (1, p(2)(n)] and existence theorems for p > p2(n), n = 2, 3. In this way we find the critical exponent for small data solutions to this problem. Our results lead to the conjecture p2(n) = p(0)(n +2) for n > 2, where p0(n) is the Strauss exponent for the classical semilinear wave equation with power nonlinearity. (C) 2015 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 13/15140-2 - Decay estimates for semilinear hyperbolic equations
• Grantee: Marcello Dabbicco
• Support Opportunities: Research Grants - Young Investigators Grants
• FAPESP's process: 14/02713-7 - Decay estimates for semilinear hyperbolic equations
• Grantee: Marcello Dabbicco
• Support Opportunities: Scholarships in Brazil - Young Researchers