APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS IN NONLINEAR SECOND ORDER EQUATIONS
Main Article Content
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.
Downloads
Metrics
Article Details
You are free to:
- Share — copy and redistribute the material in any medium or format for any purpose, even commercially.
- Adapt — remix, transform, and build upon the material for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.
Under the following terms:
- Attribution — You must give appropriate credit , provide a link to the license, and indicate if changes were made . You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
Notices:
You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation .
No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material.
