Existence of solutions for a class of IBVP for nonlinear hyperbolic equations
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ORIGINAL PAPER
Existence of solutions for a class of IBVP for nonlinear hyperbolic equations Svetlin Georgiev Georgiev1,2 • Mohamed Majdoub1,2 Received: 10 February 2020 / Accepted: 24 July 2020 / Published online: 4 August 2020 Ó Springer Nature Switzerland AG 2020
Abstract We study a class of initial boundary value problems of hyperbolic type. A new topological approach is applied to prove the existence of non-negative classical solutions. The arguments are based upon a recent theoretical result. Keywords Hyperbolic equations Positive solution Fixed point Cone Sum of operators Mathematics Subject Classification 47H10 58J20 35L15
1 Introduction This paper concerns global existence of classical solutions of one-dimensional nonlinear wave equations with initial and mixed boundary conditions. More precisely, we investigate the following IBVP 8 utt uxx ¼ f ðt; x; uÞ; t 0; x 2 ½0; L; > > > < uð0; xÞ ¼ u ðxÞ; x 2 ½0; L; 0 ð1:1Þ > > > ut ð0; xÞ ¼ u1 ðxÞ; x 2 ½0; L; : uðt; 0Þ ¼ ux ðt; LÞ ¼ 0; t 0; where L [ 0, f : ½0; 1Þ ½0; L R ! R is continuous and u0 2 C2 ð½0; LÞ, u1 2 C1 ð½0; LÞ are the initial data.
This article is part of the section ‘‘Theory of PDEs’’ edited by Eduardo Teixeira. & Svetlin Georgiev Georgiev [email protected] Mohamed Majdoub [email protected] 1
Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam, Saudi Arabia
2
Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441 Dammam, Saudi Arabia SN Partial Differential Equations and Applications
22 Page 2 of 20
SN Partial Differ. Equ. Appl. (2020) 1:22
Mixed boundary value problems arise in several areas of applied mathematics and physics, such as gas dynamics, nuclear physics, chemical reaction, studies of atomic structures, and atomic calculation. Therefore, mixed problems have attracted much interest and have been studied by many authors. See [1, 5, 6] and references cited therein. In our case, the Eq. (1.1) describes the interaction of solitary waves in elastic rods, the dynamics of one-dimensional internal gravity waves in an incompressible stratified fluid. In [13] the author investigate the following mixed problem 8 f ðuÞ; x [ 0; t [ 0; > < utt uxx ¼ ð1:2Þ 0; x ¼ 0; t [ 0; ux þ cut ¼ > : u ¼ w0 ; ut ¼ w1 ; x [ 0; t ¼ 0; where jcj 1, under the compatibility conditions w00 ð0Þ þ cw1 ð0Þ ¼ 0; w01 ð0Þ þ cðw000 ð0Þ þ Fðw0 ð0ÞÞÞ ¼ 0; 00 w000 0 ð0Þ þ cw1 ð0Þ ¼ 0:
If f 2 C1 ðRÞ, c 6¼ 1, wj 2 C2j ð½0; 1ÞÞ, j ¼ 0; 1, it is proved that there exists an open neighborhood U of f0g ½0; 1Þ such that (1.2) has exactly one solution u 2 C2 ðUÞ. Moreover, if f 2 C2 ðRÞ, wj 2 C3j ð½0; 1ÞÞ for j ¼ 0; 1, then there exists an open neighborhood U of f0g ½0; 1Þ such that (1.2) has exactly one solution u 2 C3 ðUÞ. The method used in [13] is mainly based on conservation laws. The following mixed problem is investigated in [15] 8 ¼ f ðx; tÞ; ðx; tÞ 2 ð0; LÞ ð0; TÞ; utt
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