Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Systems of nonlinear wave equations with damping and supercritical boundary and interior sources


Authors: Yanqiu Guo and Mohammad A. Rammaha
Journal: Trans. Amer. Math. Soc. 366 (2014), 2265-2325
MSC (2010): Primary 35L05, 35L20; Secondary 58J45
DOI: https://doi.org/10.1090/S0002-9947-2014-05772-3
Published electronically: January 7, 2014
MathSciNet review: 3165639
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the local and global well-posedness of the coupled nonlinear wave equations

$\displaystyle u_{tt}-\Delta u+g_1(u_t)=f_1(u,v),$    
$\displaystyle v_{tt}-\Delta v+g_2(v_t)=f_2(u,v)$    

in a bounded domain $ \Omega \subset \mathbb{R}^n$ with Robin and Dirichlét boundary conditions on $ u$ and $ v$ respectively. The nonlinearities $ f_1(u,v)$ and $ f_2(u,v)$ have supercritical exponents representing strong sources, while $ g_1(u_t)$ and $ g_2(v_t)$ act as damping. In addition, the boundary condition also contains a nonlinear source and a damping term. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35L05, 35L20, 58J45

Retrieve articles in all journals with MSC (2010): 35L05, 35L20, 58J45


Additional Information

Yanqiu Guo
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
Address at time of publication: Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: s-yguo2@math.unl.edu, yanqiu.guo@weizmann.ac.il

Mohammad A. Rammaha
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
Email: mrammaha1@math.unl.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05772-3
Keywords: Wave equations, damping and source terms, weak solutions, energy identity, nonlinear semigroups, monotone operators
Received by editor(s): July 4, 2011
Received by editor(s) in revised form: December 5, 2011
Published electronically: January 7, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society