Global existence for a class of semi-linear wave equations with variable coefficients

• Grant number: 12/19085-3
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: January 15, 2013
• End date: February 14, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Marcelo Rempel Ebert
• Grantee: Marcelo Rempel Ebert
• Visiting researcher: Marcello Dabbicco
• Visiting researcher institution: Università degli Studi di Bari - Aldo Moro, Italy
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil

Abstract

It is well known in the literature of nonlinear ordinary differential equations that the solution of the Cauchy problem blows up in finite time, even for small initial data. Recently, K. Yagdjian observed using the Floquet theory, that oscillations in the coefficients may have a negative influence on the existence of global solutions. More precisely, it has proved that the solution of the Cauchy problem blows up in finite time for a class of semi-linear wave equations with oscillating coefficient, even for small initial data. An analogous phenomenon occurs if the speed of variation decreases to zero when the time variable tends to infinity. Thereafter, were presented by several authors’ sufficient conditions for the existence of global solutions for this class of semi-linear wave equations, by assuming data with sufficiently small norms. These conditions are similar to those assumed in the work of M. Reissig and K. Yagdjian to obtain energy's estimates known as Strichartz-type decay Estimates for the wave equation with variable coefficients. One question arises naturally in problems of existence of global solutions: What about the influence of a dissipative term in these semi-linear problems? (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

D'ABBICCO, M.; EBERT, M. R.. Diffusion phenomena for the wave equation with structural damping in the L-p - L-q framework. Journal of Differential Equations, v. 256, n. 7, p. 2307-2336, APR 1 2014. (12/19085-3)

D'ABBICCO, M.; EBERT, M. R.. An application of L-p - L-q decay estimates to the semi-linear wave equation with parabolic-like structural damping. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, v. 99, p. 16-34, APR 2014. (12/19085-3)