The beam equation with nonlinear memory

Author(s)

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

Affiliation

^[1] Univ Sao Paulo, Dept Comp & Matemat, FFCLRP, Av Bandeirantes 3900, BR-14040901 Ribeirao Preto, SP - Brazil

^[2] Univ Bari, Dept Math, Via E Orabona 4, I-70125 Bari - Italy

• Document type: Journal article
• Source: ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK; v. 67, n. 3 JUN 2016.
• Web of Science Citations: 2

Abstract

In this paper, we study the critical exponent for the beam equation with nonlinear memory, i.e., u(tt) + Delta(2)u = F(t, u), where F = integral(t)(0) f(t - s)N(u)(s, x) ds, N(u) approximate to vertical bar u vertical bar(p). For suitable f and p, we prove the existence of local-in-time solutions and small data global solutions to the Cauchy problem, in homogeneous and nonhomogeneous Sobolev spaces. In some cases, we prove that the local solution cannot be extended to a global one. We also consider the limit case of power nonlinearity, i.e., F = N(u). (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