Affiliations: The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, USA
Note: [] Corresponding author: Aimin Huang, The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Rawles Hall, Bloomington, IN 47405, USA. E-mail: [email protected]
Abstract: In this article, our goal is to prove the existence and uniqueness of solution for 1D and 2D semi-linear hyperbolic equations in a bounded domain with a monotone nonlinear term. We use elliptic regularization and a finite difference scheme in time to build the approximate solutions for the semi-linear hyperbolic equations, and we utilize the regularization method together with the monotonicity and convexity of the nonlinear term to show the existence of the resulting stationary problems. Finally, the existence of the solution for the evolution problem is done by studying the convergence of the approximate solutions and by using the standard Minty method, and the uniqueness is achieved.
Keywords: semi-linear hyperbolic, nonlinear monotonicity, existence and uniqueness, integration by parts, polynomial growth
