Large time behavior of solutions for a system of nonlinear damped wave equations
This work is concerned with semilinear wave equations having linear dampings in the \(n\)-dimensional Euclidean space. The dependent variable is vector-valued with \(m\) components, \(m\) is assumed to be no less than 2. Each component of the nonlinear coupling function depending only on the dependent variable is bounded above by a power-type function prescribed by the product of the absolute values of the components of the dependent variable with given exponents. A similar bound does exist for the absolute value of the difference of of coupling functions for two different arguments. Initial values of the dependent variable and its time derivative at \(t=0\) are prescribed vector-valued functions. The authors show for \(n=1,2,3\) that if the sum of the exponents satisfies a certain inequality and the sum of the norms of initial values in appropriate Sobolev and Lebesgue spaces is sufficiently small, then there exists a unique global solution satisfying an optimal uniform decay rate. An asymptotic profile for large time is also obtained. The proof is based on the previously obtained decomposition of fundamental solutions of the linear damped wave equation into dissipative and hyperbolic parts.
scientific article; zbMATH DE number 5961149