ABSTRACT
In this section we investigate the Cauchy problem u t t − u x x = f ( t , x , u , u t , u x ) in ( 0 , ∞ ) × R , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/math5_1.jpg"/> u ( 0 , x ) = ϕ ( x ) , u t ( 0 , x ) = ψ ( x ) in R , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/math5_2.jpg"/> where ϕ , ψ ∈ C 2 ( R ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/inline-math5_1.jpg"/> , f : [ 0 , ∞ ) × R × R × R × R → R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/inline-math5_2.jpg"/> is a given continuous function, u : [ 0 , ∞ ) × R → R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/inline-math5_3.jpg"/> is unknown. We will start with the following useful lemma.
