APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS IN NONLINEAR SECOND ORDER EQUATIONS

Abstract

In the present paper there is a detailed study for the abstract second order (in time) semi linear
differential equation on Cauchy problem
in which A and B are (generally unbounded) operators which are
linear in a Banach space. This sort of problems rise up frequently with in the study of PDE (partial
differential equations). In regular we manage a nonlinear perturbation (likely related to spatial
derivatives) through the linear terms, which incorporate higher order spatial derivatives. But opposite
to the same old approach of lowering the hassle to a 1st order device in some “energy” norm space, we
make the use of the factoring approach. The approach lets in the equation to be written as integral
equation which consists a double integral related to the non linearity, reflecting the truth that the
equation is 2nd order. There are numerous blessings to this technique with a purpose to be illustrated
completely with in the examples. At first, we display that the equation is domestically posed well if the
nonlinearity is going to satisfy the local Lipschitz condition
(on soaking up linear terms into f, higher order derivatives of the operators A and B want be retained.)
For partial differential equations, this offers a great rule of thumb for figuring out if a positive hassle is
locally properly posed. Energy strategies may be used to expose worldwide existence. Secondly, this
technique applies to problems of each hyperbolic and parabolic.