Integral type Cauchy problem for abstract wave equations and applications

Abstract

We consider the integral type problem for linear and nonlinear abstract wave equations with operator coefficient in E-valued $L^p$, $1\le p\le \mathrm{\infty }$ spaces. The equation includes an abstract linear operator A that defined in a Banach space E. Here, the existence, uniqueness, uniform estimates of solution and quality properties of integral wave problem are derived. We also obtain the same properties for solutions of mixed integral problem for nonlocal and anisotropic classes, by choosing the space E and operator A.

Keywords:

• Wave equations
• semigroups of operators
• hyperbolic-operator equations
• cosine operator functions
• operator valued $L^p-$Fourier multipliers

AMS:

• 35Axx
• 47Axx
• 35Lxx
• 47Dxx

Disclosure statement

No potential conflict of interest was reported by the author(s).