# Characterization of blow-up solutions to certain nonlinear Volterra integral equations

#### Abstract

The purpose of this investigation is to determine the possibility of a blow-up solution to a particular class of wave equations with a concentrated source. That is, we are looking for solutions that become unbounded in some finite time. These wave equations are examined in the context of a nonlinear Volterra integral equation. In general, the Volterra equation is of the form (1) $v(t) = \int\sbsp{0}{t}k(t - s)g\lbrack v(s) + h(s)\rbrack ds,\ t \ge 0.$ The solution, v, appears nonlinearly in the integrand as part of the real, nonnegative function g. The difference kernel k(t, s) = $k(t - s)$ and the function h are also noted to be real and nonnegative.^ The nature of wave propagation involves vibrations or oscillations of a dependent variable. For the nonhomogeneous types of these problems, the literature has only treated source terms that are smooth and well behaved. Physical situations arise, however, when one encounters source terms that are spatially localized. Such idealized sources are characterized as impulsive or concentrated. For example, in acoustics we represent impulsive forces and pressures in sound waves in air or liquid with this type of source term. In electrostatics, we come across this scenario when we have stress waves characterized with point charges, dipoles and multipoles, resulting in such consequences as earthquakes.^ In general, nonlinear integral equations can not be solved exactly. In order to learn as much as possible about equation (1), a combination of asymptotic and functional analytic techniques are employed. This analysis assists in the characterization of conditions that may lead to solution blow-up. Once conditions for blow-up are established, analytic estimates for the critical blow-up time are sought. Research techniques include contraction mappings and contradiction arguments for determining upper and lower bounds for the existence of a solution. A characterization of the asymptotic behavior of the solution in the limit as the independent variable approaches the critical blow-up value provides additional information for understanding the behavior of these equations. Analytic techniques developed by Roberts and Olmstead (1992) are adapted to address the objectives of this study. ^

#### Subject Area

Mathematics|Physics, General

#### Recommended Citation

Kelly Molkenthin Fuller, "Characterization of blow-up solutions to certain nonlinear Volterra integral equations" (1997). Dissertations and Master's Theses (Campus Access). Paper AAI9723556.
http://digitalcommons.uri.edu/dissertations/AAI9723556

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